Identities involving generalized derivations act as Jordan homomorphisms

Abstract

Let R be a prime ring with char(R) is not equal to 2 and $\pi (\omega _1,\ldots ,\omega _n)$ be a noncentral multilinear polynomial over the extended centroid C of R. If $F_1$, $F_2$ and $F_3$ are generalized derivations on R such that $F_1(F_3(\xi ^2))=F_2(\xi )F_3(\xi )$ for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$, then we describe all possible forms of generalized derivations $F_1$, $F_2$ and $F_3$.

1 Introduction

Let R be an associative ring. A mapping $\zeta :R\rightarrow R$ is said to be a derivation on R if

$$\begin{aligned} \zeta (u_1+u_2)=\zeta (u_1)+\zeta (u_2) \text {, } \zeta (u_1u_2)=\zeta (u_1)u_2+u_1\zeta (u_2), \end{aligned}$$

for all $u_1, u_2\in R$. The commutator of $u_1$ and $u_2$ is denoted by $[u_1, u_2]=u_1u_2-u_2u_1$ for $u_1, u_2 \in R$, which is called the Lie commutator of $u_1$ and $u_2$. For fix $a\in R$, define a mapping $g_a:R \rightarrow R$ by $g_a (u)=[a, u]$ for all $u\in R$. We can easily prove that $g_a$ is a derivation on R and usually it is called an inner derivation on R. A mapping $\zeta : R\rightarrow R$ is said to be a generalized derivation if there exists a derivation g on R such that

$$\begin{aligned} \zeta (u_1+u_2)=\zeta (u_1)+\zeta (u_2) \text {, } \zeta (u_1u_2)=\zeta (u_1)u_2+u_1g(u_2), \end{aligned}$$

for all $u_1, u_2\in R$ (for more details see [4]). Let $s_1, s_2$ be fixed elements in R and $\zeta _{(s_1, s_2)}: R\rightarrow R$ be a mapping defined by $\zeta _{(s_1, s_2)}(u)=s_1u+us_2$ for all $u\in R$. Here, we can easily prove that $\zeta _{(s_1, s_2)}$ is a generalized derivation on R and it is called a generalized inner derivation on R.

A mapping $\zeta :R\rightarrow R$ is called homomorphism and anti-homomorphism if

$$\begin{aligned} \zeta (u_1+u_2)=\zeta (u_1)+\zeta (u_2) \text {, } \zeta (u_1u_2)=\zeta (u_1)\zeta (u_2) \end{aligned}$$

and

$$\begin{aligned} \zeta (u_1+u_2)=\zeta (u_1)+\zeta (u_2) \text {, } \zeta (u_1u_2)=\zeta (u_2)\zeta (u_1), \end{aligned}$$

for all $u_1, u_2\in R$ respectively. A mapping $\zeta :R\rightarrow R$ is called Jordan homomorphism if

$$\begin{aligned} \zeta (u_1+u_2)=\zeta (u_1)+\zeta (u_2) \text {, } \zeta (u^2)=\zeta (u)^2, \end{aligned}$$

for all $u, u_1, u_2\in R$. We notice that every homomorphism and anti-homomorphism is a Jordan homomorphism but the converse is not true in general. The following example justify our observation.

Example 1.1

Suppose that $*$ is an involution on a ring R and is a ring with the properties $t_1 z t_2=0$, for all $t_1, t_2\in R$, where $z\in Z(R)$. We define a function $\zeta : S\rightarrow S$ such that $\zeta (t_1, t_2)=(zt_1, t_2^{*})$, for all $t_1, t_2 \in R$. It is clear that $\zeta$ is a Jordan homomorphism on S but not a homomorphism on S.

In 1956, Herstein [20] proved that every Jordan homomorphism from a ring R onto a prime ring $R'$ with char$(R)\ne 2, 3$ is either a homomorphism or anti-homomorphism. Further, in 1957 Smiley [28] extended the Herstein’s result [20] and proved that the statement of the Herstein’s result is still true without taking the characteristic is not equal to 3.

In the literature, several mathematicians describe the structure of prime ring R with the additive mappings which acts as a homomorphism or anti-homomorphism or Jordan homomorphism on Lie ideals, Jordan ideals or some appropriate subsets of R. In this line of investigation, Bell and Kappe [6] proved the first result in the context of derivation. More precisely, they proved that there is no nonzero derivation on prime ring R which acts as a homomorphism or anti-homomorphism on nonzero right ideal of R. Further, above [6] result was extended by Wang and You [33] to Lie ideal case under suitable restriction.

Recently, generalized derivation which behaves as a Jordan homomorphism, Lie homomorphism, homomorphism, anti-homomorphism were discuss in [11, 12, 16,17,18, 29,30,31], where further references can be found.

On the other hand, Posner [27] gave a remarkable results concerning centralizing mapping on prime ring R. More specifically, Posner [27] proved that if d is a nonzero centralizing derivation on prime ring R, then R must be commutative. Further, Posner’s [27] result was extended by Bre$\check{s}$ar [5]. Later on many mathematicians studied the structure of prime rings as well as structure of additive mappings which behaves as a commuting or centralizing mappings. In this line of investigation, the readers are refer to ( [1, 2, 29, 32], where further references can be found).

Carini et al. in [8] considered a noncommutative prime ring R of characteristic different from 2 with Utumi quotient ring U and extended centroid C, $\pi (\omega _1,\ldots ,\omega _n)$ a multilinear polynomial over C which is not an identity for R, F and G two nonzero generalized derivations of R. If $F(\xi )G(\xi )=0$ for all $\xi \in \pi (R)=\{\pi (\omega _1,\ldots ,\omega _n)\mid \omega _i\in R\}$, then they gave the complete possible forms of F and G.

Throughout the following U denotes the Utumi quotient ring of prime ring R, C denotes the center of U which is called the extended centroid of R and $\pi (\omega _1,\ldots ,\omega _n)$ is noncentral multilinear polynomial over C. More details about U and C readers can found in [3] and [9].

Motivated by above cited results, our result is the following.

2 Results

Theorem 2.1

Suppose that R is a prime ring of characteristic not equal to 2 and G, F, H are three generalized derivations on R. If

$$\begin{aligned} G(H(\xi ^2))=F(\xi )H(\xi ) \end{aligned}$$

for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$, then one of the following holds:

• (i) $H=0$;

• (ii) there exist $\sigma \in C$, $w_1 \in U$ such that $G(r)=F(r)=w_1 r$ and $H(r)=\sigma r$ for all $r\in R$;

• (iii) there exist $w_1, w_2, w_3 \in U$, $\sigma \in C$ such that $H(r)=r w_1$, $F(r)=w_2 r$ and $G(r)=\sigma r+w_2 r+ r w_3$ with $w_1 w_3=-\sigma w_1$;

• (iv) there exist $w_1, w_2, w_3 \in U$ such that $H(r)=w_1 r$, $F(r)=r w_2$ and $G(r)=w_3 r$ for all $r\in R$ with $w_3 w_1=w_2 w_1=\alpha \in C$;

• (v) $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and one of the following holds:

  • (a) there exist $0\ne \sigma \in C$, $w_1, w_2 \in U$ such that $H(r)=\sigma r$, $F(r)=w_1 r$ and $G(r)=[w_2, r]+r w_1$ for all $r\in R$;

  • (b) there exist $w_1, w_2, w_3, w_4 \in U$, $\alpha \in C$ such that $G(r)=w_3 r+r w_4$ and either $H(r)=r w_1$, $F(r)=w_2 r$ with $w_3 w_1+w_1 w_4=w_2 w_1$ or $H(r)=w_1 r$, $F(r)=r w_2$ for all $r\in R$ with $w_3 w_1 +w_1 w_4=w_2 w_1 =\alpha \in C$;

  • (c) there exist $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ with $w_3(w_1+w_2)+(w_1+w_2)w_4=0$.

• (vi) there exist $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ and R satisfies $s_4$;

• (vii) $G=0$ and $F=0$;

• (viii) there exist $\sigma _1, \sigma _2, \sigma _3 \in C$, $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ with $w_4+ \sigma _3 w_2=\sigma _1$, $w_3-\sigma _3 w_1=\sigma _2$ and $w_3 w_1+\sigma _1 w_1=-(w_2 w_4+\sigma _2 w_2) \in C$.

In particular for $G=I$, the identity mapping and $F=H$ in the Theorem 2.1, we get the following result.

Corollary 2.2

Suppose that R is a prime ring of characteristic not equal to 2 and H is a generalized derivation on R. If $H(\xi ^2)=H(\xi )^2$ for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$, then either $H=0$ or $H(t)=t$ for all $t\in R$.

Corollary 2.3

( [8], Main theorem) By taking $G=0$ in Theorem 2.1, we get the Carini, De Filippis and Gsudo result.

In particular for $H=I$, the identity mapping in Theorem 2.1, we get the following.

Corollary 2.4

Suppose that R is a prime ring of characteristic not equal to 2 and G, F are generalized derivations on R. If $G(\xi ^2)=F(\xi )\xi$ for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$, then one of the following holds:

• (i) $G=0$ and $F=0$;

• (ii) there exists $w_1\in U$ with $F(t)=G(t)=w_1 t$ for all $t\in R$;

• (iii) there exist $w_1, w_2 \in U$ with $F(t)=w_1 t$, $G(t)=[w_2, t]+tw_1$ for all $t\in R$ and $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R;

• (iv) there exist $w_1, w_2\in U$ with $G(t)=w_1 t+t w_2$, $F=0$ and R satisfies $s_4$.

Since difference of two generalized derivations is a generalized derivation on ring R. By substituting $G=0$ and $F=F-I$, where I is the identity mapping on R in Theorem 2.1, we have the following.

Corollary 2.5

Suppose that R is a prime ring of characteristic not equal to 2 and $H\ne 0$, F are generalized derivations on R. If

$$\begin{aligned} F(\xi )H(\xi )-\xi H(\xi )=0, \end{aligned}$$

for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$, then one of the following holds:

• (i) $F=I$, the identity mapping;

• (ii) there exist $w_1, w_2\in U$ with $F(t)=t(w_1+1)$, $H(t)=w_2 t$ for all $t\in R$ with $w_1 w_2=0$;

• (iii) there exist $w_1, w_2\in U$ with $F(t)=(w_1+1)t$, $H(t)=tw_2$ for all $t\in R$ with $w_1 w_2=0$ and $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R.

3 Preliminaries

Let h and $\delta$ be two derivations on R. We denote by $\pi ^h(\omega _1,\ldots ,\omega _n)$ the polynomials obtained from $\pi (\omega _1,\ldots ,\omega _n)$ replacing each coefficients $\alpha _\sigma$ with $h(\alpha _\sigma )$. Then we have

$$\begin{aligned} h(\pi (\omega _1,\ldots ,\omega _n))=\pi ^h(\omega _1,\ldots ,\omega _n)+\sum \limits _i\pi (\omega _1,\ldots ,h(\omega _i),\ldots ,\omega _n) \end{aligned}$$

and

$$\begin{aligned} h\delta (\pi (\omega _1,\ldots ,\omega _n))= & {} \pi ^{h \delta }(\omega _1,\ldots ,\omega _n)+\sum \limits _i \pi ^h(\omega _1,\ldots ,\delta (\omega _i),\ldots ,\omega _n)\nonumber \\+ & {} \sum \limits _i \pi ^{\delta }(\omega _1,\ldots ,h(\omega _i),\ldots ,\omega _n)+\sum \limits _i \pi (\omega _1, \ldots , h\delta (\omega _i), \ldots , \omega _n)\nonumber \\+ & {} \sum \limits _{i\ne j}\pi (\omega _1, \ldots , h(\omega _i), \ldots , \delta (\omega _j), \ldots , \omega _n). \end{aligned}$$

(1)

The following results are frequently used to prove our theorem. Let R be a prime ring and I be a two sided ideal of R.

Remark 3.1

By ([9]), R, I and U satisfy the same generalized polynomial identities with coefficients in U.

Remark 3.2

By ([24]), R, I and U satisfy the same differential identities.

4 G, F and H are inner maps

This section deals the case when G, F and H all are generalized inner derivations. Suppose $G(x)=p_{5}x+xp_{6}$, $F(x)=p_{1}x+xp_{2}$ and $H(x)=p_{3}x+xp_{4}$ for all $x\in R$, where $p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}\in U$. From the hypothesis $G(H(\pi (\xi )^2))=F(\pi (\xi ))H(\pi (\xi ))$ we get the expression $p_{5}\Big (p_{3}X^2+X^2 p_{4}\Big )+\Big (p_{3}X^2+X^2 p_{4}\Big )p_{6}=\Big (p_{1}X+Xp_{2}\Big )\Big (p_{3}X+Xp_{4}\Big )$ that is $a_1 X^2+a_2X^2a_3+a_4X^2a_5-a_6Xa_4X-Xa_7X-a_6X^2a_3-Xa_8Xa_3+X^2a_9=0$ for all $X=\pi (\omega _1,\ldots ,\omega _n)$, where $a_1=p_{5}p_{3}, a_2=p_5, a_3=p_4, a_4=p_3, a_5=p_6, a_6=p_1, a_7=p_2 p_3, a_8=p_2, a_9=p_4 p_6$.

Proposition 4.1

Suppose R is a prime ring with characteristic is not equal to 2. If H, G, F are three generalized inner derivations on R such that

$$\begin{aligned} G(H(\xi ^2))=F(\xi )H(\xi ), \end{aligned}$$

for all $\xi =\pi (\omega _1,\ldots ,\omega _n)$, $\omega _1,\ldots ,\omega _n \in R$. Then one of the following holds:

• (i) $H=0$;

• (ii) there exist $\sigma \in C$, $w_1\in U$ with $G(r)=F(r)=w_1 r$ and $H(r)=\sigma r$ for all $r\in R$;

• (iii) there exist $w_1, w_2, w_3 \in U$, $\sigma \in C$ with $H(r)=rw_1$, $F(r)=w_2 r$ and $G(r)=\sigma r+w_2 r+r w_3$ with $w_1 w_3=-\sigma w_1$;

• (iv) there exist $w_1, w_2, w_3 \in U$ with $H(r)=w_1 r$, $F(r)=r w_2$ and $G(r)=w_3 r$ for all $r\in R$ with $w_3 w_1=w_2 w_1=\alpha \in C$;

• (v) $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and one of the following holds:

  • (a) there exist $0\ne \sigma \in C$, $w_1, w_2 \in U$ with $H(r)=\sigma r$, $F(r)=w_1 r$ and $G(r)=[w_2, r]+r w_1$ for all $r\in R$;

  • (b) there exist $w_1, w_2, w_3, w_4 \in U$, $\alpha \in C$ such that $G(r)=w_3 r+r w_4$ and either $H(r)=r w_1$, $F(r)=w_2 r$ with $w_3 w_1+w_1 w_4=w_2 w_1$ or $H(r)=w_1 r$, $F(r)=r w_2$ with $w_3 w_1 +w_1 w_4=w_2 w_1=\alpha \in C$;

  • (c) there exist $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ with $w_3(w_1+w_2)+(w_1+w_2)w_4=0$.

• (vi) there exist $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ and R satisfies $s_4$;

• (vii) $G=0$ and $F=0$;

• (viii) there exist $\sigma _1, \sigma _2, \sigma _3 \in C$, $w_1, w_2, w_3, w_4 \in U$ such that $G(r)=w_3 r+r w_4$, $H(r)=w_1 r+r w_2$, $F=0$ with $w_4+\sigma _3 w_2=\sigma _1$, $w_3-\sigma _3 w_1=\sigma _2$ and $w_3 w_1+\sigma _1 w_1=-(w_2 w_4+ \sigma _2 w_2) \in C$.

We need the following results to prove the above proposition.

Lemma 4.2

[15, Lemma 1] Let $m\ge 2$ and K be an infinite field. Let $A_1,A_2,\ldots ,A_n$ are non scalar matrices in $M_m(K)$ then there exists an invertible matrix $N\in M_m(K)$ such that all matrices $NA_1N^{-1}$, $NA_2N^{-1}$, $\ldots$, $NA_nN^{-1}$ have all nonzero entries.

Proposition 4.3

Let $R=M_m(K)$ be the ring of all $m\times m$ matrices over the field K with characteristic not equal to 2 and $m\ge 2$. Let $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, $q_6$, $q_7$, $q_8$, $q_9\in R$ such that $q_1 X^2+q_2X^2q_3+q_4X^2q_5-q_6Xq_4X-Xq_7X-q_6X^2q_3-Xq_8Xq_3+X^2q_9=0$ for all $X=\pi (\omega _1,\ldots ,\omega _n)\in f(R)$, then one of the following holds:

• (i) $q_4, q_3$ are central;

• (ii) $q_4, q_8$ are central;

• (iii) $q_6, q_3$ are central;

• (iv) $q_6, q_8$ are central.

Proof

We shall prove this by contradiction. Suppose that $q_4\notin Z(R)$ and $q_6\notin Z(R)$. By the hypothesis, R satisfies the generalized polynomial identity

$$\begin{aligned}&q_1 X^2+q_2X^2q_3+q_4X^2q_5-q_6Xq_4X-Xq_7X-q_6X^2q_3\nonumber \\&-Xq_8Xq_3+X^2q_9=0, \end{aligned}$$

(2)

for all $X\in \pi (R^n)$. We have the following cases.

Case-I: If K is infinite, then by Lemma 4.2 there exists a K-automorphism $\phi$ of $M_m(K)$ such that $q_4'=\phi (q_4)$, $q_6'=\phi (q_6)$ have all nonzero entries. Clearly $q_4'$, $q_6'$, $q_1'=\phi (q_1)$, $q_2'=\phi (q_2)$, $q_3'=\phi (q_3)$, $q_5'=\phi (q_5)$, $q_7'=\phi (q_7)$, $q_8'=\phi (q_8)$ and $q_9'=\phi (q_9)$ must satisfy the relation (2). Now we can replace $q_1$, $q_2$, $q_3$, $q_4$, $q_5$, $q_6$, $q_7$, $q_8$, $q_9$ with $q_1'$, $q_2'$, $q_3'$, $q_4'$, $q_5'$, $q_6'$, $q_7'$, $q_8'$, $q_9'$ respectively.

Let $e_{ij}$ be the matrix such that (i, j)-entry is 1 and remaining other entries are zero. It is given $X=\pi (\omega _1,\ldots ,\omega _n)$ is not central, by [24] (see also [25]), there exist $\omega _1, \ldots , \omega _n \in M_m(K)$ and $\gamma \in K{\setminus }\{0\}$ such that $X=\pi (\omega _1,\ldots ,\omega _n)=\gamma e_{st}$, with $s \ne t$. Moreover, since the set $\{X=\pi (\omega _1,\ldots ,\omega _n): \omega _1, \ldots , \omega _n \in M_m(K)\}$ is invariant under the action of all K-automorphisms of $M_m(K)$, then $i \ne j$ there exist $\omega _1, \ldots , \omega _n \in M_m(K)$ such that $X=\pi (\omega _1,\ldots ,\omega _n) =e_{ij}$. Hence by (2) we have

$$\begin{aligned}&-q_6e_{ij}q_4e_{ij}-e_{ij}q_7e_{ij}-e_{ij}q_8e_{ij}q_3=0. \end{aligned}$$

(3)

Left multiplying above relation by $e_{ij}$, we obtain $e_{ij}q_6e_{ij}q_4e_{ij}=0$. It implies that ${q_6}_{ij}{q_4}_{ij} e_{ij}=0$. This implies that either ${q_6}_{ij}=0$ or ${q_4}_{ij}=0$, a contradiction and thus we get either $q_4$ is central or $q_6$ is central.

In the same fashion, we can show that either $q_3$ is central or $q_8$ is central. Combining these two results, we get our conclusions.

Case-II: Let K be a finite. Suppose F is an infinite extension of K. Suppose that ${\overline{R}}=M_m(F)\cong R\otimes _K F$. Note that $\pi (\omega _1,\ldots ,\omega _n)$ is central valued on R if and only if it is central valued on ${\overline{R}}$. Suppose ${\textsf{Q}}(\omega _1,\ldots ,\omega _n)$ is the generalized polynomial such that

$$\begin{aligned} {\textsf{Q}}(\omega _1,\ldots ,\omega _n)= & {} q_1 X^2+q_2X^2q_3+q_4X^2q_5-q_6Xq_4X-Xq_7X\\{} & {} -q_6X^2q_3-Xq_8Xq_3+X^2q_9 \end{aligned}$$

is a GPI for R.

Since ${\textsf{Q}}(\omega _1,\ldots ,\omega _n)$ is a multihomogeneous of multidegree $(2,\ldots ,2)$ in the indeterminates $\omega _1, \ldots , \omega _n$. By complete linearization of ${\textsf{Q}}(\omega _1, \ldots , \omega _n)$ we get a multilinear generalized polynomial $\Theta (\omega _1, \ldots , \omega _n, x_1, \ldots , x_n)$ in 2n indeterminates, moreover

$$\begin{aligned} \Theta (\omega _1, \ldots , \omega _n, \omega _1, \ldots , \omega _n)=2^n {\textsf{Q}}(\omega _1, \ldots , \omega _n). \end{aligned}$$

It is clear that the multilinear polynomial $\Theta (\omega _1, \ldots , \omega _n, x_1, \ldots , x_n)$ is a generalized polynomial identity for R and ${\overline{R}}$. By assumption char$(R)\ne 2$ we obtain ${\textsf{Q}}(\omega _1,\ldots ,\omega _n)=0$ for all $\omega _1,\ldots ,\omega _n \in {\overline{R}}$ and then we get a contradiction from Case-I. Thus we get $q_4\in C$ or $q_6\in C$.

Similarly, we can prove that either $q_3$ is central or $q_8$ is central. $\square$

Lemma 4.4

Let R be a prime ring of characteristic not equal to 2. If $q_1, q_2, q_3, q_4, q_5, q_6, q_7, q_8, q_9\in R$ such that $q_1 X^2+q_2X^2q_3+q_4X^2q_5-q_6Xq_4X-Xq_7X-q_6X^2q_3-Xq_8Xq_3+X^2q_9=0$ for all $X=\pi (\xi )$, $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$. Then one of the following holds:

• (i) $q_4, q_3$ are central;

• (ii) $q_4, q_8$ are central;

• (iii) $q_6, q_3$ are central;

• (iv) $q_6, q_8$ are central.

Proof

First, we will show that one of $q_4$ or $q_6$ is central. On contrary suppose that both $q_4$ and $q_6$ both are not central. By hypothesis, we have

$$\begin{aligned}&\textrm{h}(\omega _1,\ldots ,\omega _n)=q_1 \pi (\omega _1,\ldots ,\omega _n)^2+q_2\pi (\omega _1,\ldots ,\omega _n)^2q_3\\&\quad +q_4\pi (\omega _1,\ldots ,\omega _n)^2q_5-q_6\pi (\omega _1,\ldots ,\omega _n)q_4\pi (\omega _1,\ldots ,\omega _n)\\&\quad -\pi (\omega _1,\ldots ,\omega _n)q_7\pi (\omega _1,\ldots ,\omega _n)-q_6\pi (\omega _1,\ldots ,\omega _n)^2q_3\\&\quad -\pi (\omega _1,\ldots ,\omega _n)q_8\pi (\omega _1,\ldots ,\omega _n)q_3+\pi (\omega _1,\ldots ,\omega _n)^2q_9, \end{aligned}$$

for all $\omega _1,\ldots ,\omega _n \in R$. By Remark 3.1, R and U satisfy same generalized polynomial identity (GPI). Then U satisfies $\textrm{h}(\omega _1,\ldots ,\omega _n)=0.$ Let $\textrm{h}(\omega _1,\ldots ,\omega _n)$ be a trivial GPI for U, T be the free product of U and $C\{\omega _1,\ldots ,\omega _n\}$, the free C-algebra in non commuting indeterminates $\omega _1,\ldots , \omega _n$ that is $T=U*_CC\{\omega _1,\ldots ,\omega _n\}$. Then, $\textrm{h}(\omega _1,\ldots ,\omega _n)=0$ in $T=U*_CC\{\omega _1,\ldots ,\omega _n\}$. The term

$$\begin{aligned}{} & {} -q_6\pi (\omega _1,\ldots ,\omega _n)q_4\pi (\omega _1,\ldots ,\omega _n)-\pi (\omega _1,\ldots ,\omega _n)q_7\pi (\omega _1,\ldots ,\omega _n)\\{} & {} -\pi (\omega _1,\ldots ,\omega _n)q_8\pi (\omega _1,\ldots ,\omega _n)q_3 \end{aligned}$$

appears nontrivially in $\textrm{h}(\omega _1,\ldots ,\omega _n)$. Since $q_6\notin C$, then we get

$$\begin{aligned} q_6\pi (\omega _1,\ldots ,\omega _n)q_4\pi (\omega _1,\ldots ,\omega _n)=0_{T}, \end{aligned}$$

gives a contradiction since neither $q_6 \in C$ nor $q_4 \in C$. Thus we get either $q_4 \in C$ or $q_6 \in C$.

Let $q_4 \in C$. Suppose that $q_3 \notin C$ and $q_8 \notin C$. Since $q_4 \in C$, U satisfies

$$\begin{aligned}&\textrm{Q}(\omega _1,\ldots ,\omega _n)=(q_1-q_6q_4) \pi (\omega _1,\ldots ,\omega _n)^2+q_2\pi (\omega _1,\ldots ,\omega _n)^2q_3\\&-\pi (\omega _1,\ldots ,\omega _n)q_7\pi (\omega _1,\ldots ,\omega _n)-q_6\pi (\omega _1,\ldots ,\omega _n)^2q_3\\&-\pi (\omega _1,\ldots ,\omega _n)q_8\pi (\omega _1,\ldots ,\omega _n)q_3+\pi (\omega _1,\ldots ,\omega _n)^2(q_9+q_4q_5). \end{aligned}$$

This is again a trivial GPI. Then $\textrm{Q}(\omega _1,\ldots ,\omega _n)=0_T$ and the term

$$\begin{aligned} \pi (\omega _1,\ldots ,\omega _n)q_8\pi (\omega _1,\ldots ,\omega _n)q_3 \end{aligned}$$

appears nontrivially in $\textrm{Q}(\omega _1,\ldots ,\omega _n)$. This gives that either $q_3$ is central or $q_8$ is central, a contradiction. Thus we conclude that either $q_4 \in C$, $q_3\in C$ or $q_4 \in C$, $q_8 \in C$.

In the same fashion, we can prove that either $q_6 \in C$, $q_3\in C$ or $q_6 \in C$, $q_8 \in C$.

Suppose $\textrm{h}(\omega _1,\ldots ,\omega _n)$ is a non trivial GPI for U. If C is infinite field, we have $\textrm{h}(\omega _1,\ldots ,\omega _n)=0$ for all $\omega _1,\ldots ,\omega _n \in U \otimes _C{\overline{C}}$, where ${\overline{C}}$ denotes the algebraic closure of C. By [13, Theorems 2.5 and 3.5], $U \otimes _C {\overline{C}}$ and U both are centrally closed and prime, then we replace R by U or $U \otimes _C {\overline{C}}$ according to C finite or infinite. Then $\textrm{h}(\omega _1,\ldots ,\omega _n)=0$ for all $\omega _1,\ldots ,\omega _n \in R$ and R is centrally closed over C. By Martindale’s theorem [26], R is then a primitive ring with nonzero socle soc(R) and with C as its associated division ring. Then, by Jacobson’s theorem [21, p.75], R is isomorphic to a dense ring of linear transformations of a vector space V over C.

If the dimension of V is finite, that is, $\dim _C V = m$. Then $R\cong M_m(C)$ (by density of R). By our hypothesis $\pi (\omega _1,\ldots ,\omega _n)$ is not central valued on R it implies that R must be noncommutative. Then we can assume that $m\ge 2$. By Proposition 4.3, it implies that either $q_4\in C$, $q_3\in C$ or $q_4\in C$, $q_8\in C$ or $q_6\in C$, $q_3 \in C$ or $q_6\in C$, $q_8 \in C$.

Next we suppose that V is infinite dimensional over C. By Martindale’s theorem [26, Theorem 3], for any $e^2=e\in soc(R)$ we have $eRe\cong M_t(C)$ with $t=\dim _CVe$. We shall prove this case by contradiction. Suppose that none of $q_4$, $q_6$, $q_3$, $q_8$ are not in the center C. Then there exist $u_1, u_2, u_3, u_4\in soc(R)$ with $[q_4,u_1]\ne 0$, $[q_3,u_3]\ne 0$, $[q_8,u_4]\ne 0$ and $[q_6,u_2]\ne 0$. By Litoff’s Theorem [14], there exists an element $e \in soc(R)$ with $e^2=e$ and $q_4 u_1, u_1q_4, q_6u_2, u_2 q_6, q_3u_3, u_3q_3, q_8u_4, u_4q_8\in eRe$. Since R satisfies generalized identity

$$\begin{aligned}{} & {} e\Big \{q_1 \pi (e\omega _1e,\ldots ,e\omega _ne)^2+q_2\pi (e\omega _1e,\ldots ,e\omega _ne)^2q_3\\{} & {} \quad +q_4\pi (e\omega _1e,\ldots ,e\omega _ne)^2q_5-q_6\pi (e\omega _1e,\ldots ,e\omega _ne)q_4\pi (e\omega _1e,\ldots ,e\omega _ne)\\{} & {} \quad -\pi (e\omega _1e,\ldots ,e\omega _ne)q_7\pi (e\omega _1e,\ldots ,e\omega _ne)-q_6\pi (e\omega _1e,\ldots ,e\omega _ne)^2q_3\\{} & {} \quad -\pi (e\omega _1e,\ldots ,e\omega _ne)q_8\pi (e\omega _1e,\ldots ,e\omega _ne)q_3+\pi (e\omega _1e,\ldots ,e\omega _ne)^2q_9\Big \}e, \end{aligned}$$

the subring eRe satisfies

$$\begin{aligned}{} & {} eq_1e \pi (\omega _1,\ldots ,\omega _n)^2+eq_2e\pi (\omega _1,\ldots ,\omega _n)^2eq_3e\\{} & {} \quad +eq_4e\pi (\omega _1,\ldots ,\omega _n)^2eq_5e-eq_6e\pi (\omega _1,\ldots ,\omega _n)eq_4e\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad -\pi (\omega _1,\ldots ,\omega _n)eq_7e\pi (\omega _1,\ldots ,\omega _n)-eq_6e\pi (\omega _1,\ldots ,\omega _n)^2eq_3e\\{} & {} \quad -\pi (\omega _1,\ldots ,\omega _n)eq_8e\pi (\omega _1,\ldots ,\omega _n)eq_3e+\pi (\omega _1,\ldots ,\omega _n)^2eq_9e. \end{aligned}$$

Then either $eq_4e$ or $eq_6$e and either $eq_3e$ or $eq_8e$ are central elements of eRe (by the above finite dimensional case). Then we get either $q_4 u_1=eq_4eu_1=u_1 eq_4 e=u_1q_4$ or $q_6 u_2=eq_6eu_2=u_2 eq_6 e=u_2q_6$ and either $q_3 u_3=eq_3eu_3=u_3 eq_3 e=u_3q_3$ or $q_8 u_4=eq_8eu_4=u_4 eq_8 e=u_4q_8$, a contradiction. $\square$

Lemma 4.5

[12, Lemma 2.9] Let R be a prime ring of characteristic not equal to 2, $q_1, q_2, q_3, q_4\in U$ and $p(\omega _1,\ldots ,\omega _n)$ be any polynomial over C which is not identity for R. If $q_1p(\xi )+p(\xi )q_2+q_3p(\xi )q_4=0$ for all $\xi \in R^n$ then one of the following conditions holds:

• (i) $q_2, q_4\in C$ and $q_1+q_2+q_3q_4=0$,

• (ii) $q_1, q_3\in C$ and $q_1+q_2+q_3q_4=0$,

• (iii) $q_1+q_2+q_3q_4=0$ and $p(\omega _1,\ldots ,\omega _n)$ is central valued on R.

Lemma 4.6

Let R be a prime ring of characteristic is not equal to 2 and $q_1, q_2, q_3, q_4 \in R$. If $q_1\pi (\xi )^2-\pi (\xi )q_2\pi (\xi )+q_3\pi (\xi )^2 q_4=0$ for all $\xi =(\omega _1, \ldots , \omega _n)\in R^n$, then one of the following holds:

• (i) $q_4\in C$ and $q_1+q_3q_4=q_2=\alpha \in C$;

• (ii) $q_1, q_3\in C$ and $q_1+q_3q_4=q_2=\alpha \in C$;

• (iii) $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and $q_1+q_3q_4=q_2=\alpha \in C$.

Proof

By using similar argument as we have used in Lemma 4.4, we get $q_2\in C$. Then our hypothesis gives that

$$\begin{aligned} (q_1-q_2)\pi (\xi )^2+q_3\pi (\xi )^2q_4=0 \end{aligned}$$

for all $\xi =(\omega _1, \ldots , \omega _n)\in R^n$. From Lemma 4.5, we have one of the following.

• $q_4\in C$ and $q_1-q_2+q_3q_4=0$, which implies that $q_1+q_3q_4=q_2=\alpha \in C$. In this case, we get $q_2, q_4\in C$ and $q_1+q_3q_4=q_2=\alpha \in C$ for some $\alpha \in C$, which is conclusion (i);

• $q_1-q_2\in C$, $q_3\in C$ and $q_1-q_2+q_3q_4=0$. Thus we get $q_1, q_2, q_3 \in C$ and $q_1+q_3q_4=q_2=\alpha \in C$ for some $\alpha \in C$, which is conclusion (ii);

• $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and $q_1+q_3q_4=q_2=\alpha \in C$ for some $\alpha \in C$, which is conclusion (iii).

$\square$

Lemma 4.7

Let R be a prime ring of characteristic not equal to 2 and $q_1, q_2, q_3, q_4, q_5, q_6, q_7, q_8\in R$. If $q_1 \pi (\xi )^2+q_2\pi (\xi )^2q_3+q_4\pi (\xi )^2q_5+\pi (\xi )^2q_6-\pi (\xi )q_7\pi (\xi )-\pi (\xi )^2q_8=0$ for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$, then $q_7\in C$.

Proof

By using similar argument as we have used in Lemma 4.4, we get our conclusion. $\square$

The following lemma is a particular case of Lemma 3 of [2].

Lemma 4.8

Let R be a noncommutative prime ring of characteristic different from 2 and $q_1, q_2, q_3 \in U$. If $\pi (\xi )q_1\pi (\xi )+\pi (\xi )^2q_2-q_3\pi (\xi )^2=0$ for all $\xi =(\omega _1,\ldots ,\omega _n) \in R^n$, then one of the following holds:

• (i) $q_2$, $q_3\in C$ and $q_3-q_2=q_1=\mu \in C$,

• (ii) $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and $q_3-q_2=q_1=\mu \in C$.

Lemma 4.9

[19, Proposition 2.13] Let R be a prime ring of characteristic is not equal to 2. Suppose there exist $a,b,c,q,u,v \in R$ such that $ax+bxc+qxu+xv=0$, for all $x\in S=[R, R]$. Then one of the following holds:

• 1. R satisfies $s_4$;

• 2. $c,q \in Z(R)$ and $a+bc=-(v+uq)\in Z(R)$;

• 3. $c,u,v \in Z(R)$ and $a+bc+qu+v=0$;

• 4. $a,b,q \in Z(R)$ and $a+bc+qu+v=0$;

• 5. $b,u \in Z(R)$ and $a+qu=-(v+bc)\in Z(R)$;

• 6. there exist $\lambda , \mu , \eta \in Z(R)$ such that $u+\eta c=\lambda$, $b-\eta q=\mu$ and $a+\lambda q=-(v+\mu c) \in Z(R)$.

Now we are ready to prove Proposition 4.1.

Proof of Proposition 4.1:

By the hypothesis, we have

$$\begin{aligned}{} & {} p_5\Big (p_3\pi (\omega _1,\ldots ,\omega _n)^2+\pi (\omega _1,\ldots ,\omega _n)^2 p_4\Big )+\Big (p_3\pi (\omega _1,\ldots ,\omega _n)^2\nonumber \\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)^2 p_4\Big )p_6=\Big (p_1\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)p_2\Big )\Big (p_3\pi (\omega _1,\ldots ,\omega _n)+\pi (\omega _1,\ldots ,\omega _n)p_4\Big ). \end{aligned}$$

(4)

That is

$$\begin{aligned}{} & {} p_5p_3\pi (\omega _1,\ldots ,\omega _n)^2+p_5\pi (\omega _1,\ldots ,\omega _n)^2 p_4+p_3\pi (r\omega _1,\ldots ,\omega _n)^2p_6\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)^2p_4p_6=p_1\pi (\omega _1,\ldots ,\omega _n)p_3\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)p_2p_3\pi (\omega _1,\ldots ,\omega _n)+p_1f(\omega _1,\ldots ,\omega _n)^2p_4\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)p_2\pi (\omega _1,\ldots ,\omega _n)p_4, \end{aligned}$$

for all $\omega _1,\cdots ,\omega _n\in R$. By Lemma 4.4 we get one of the following:

• 1. $p_3, p_4$ are central;

• 2. $p_3, p_2$ are central;

• 3. $p_1, p_4$ are central;

• 4. $p_1, p_2$ are central.

Case-I: If $p_3, p_4\in C$, then $H(t)=(p_3+p_4)t=\lambda t$ for all $t\in R$. If $\lambda =0$, then we have conclusion (i). Suppose that $\lambda \ne 0$. Then equation (4) reduces to

$$\begin{aligned} (p_5-p_1)\pi (\xi )^2+\pi (\xi )^2p_6-\pi (\xi )p_2\pi (\xi )=0 \end{aligned}$$

for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$. Then by Lemma 4.8, we have one of the following:

• $p_6\in C$, $p_5-p_1\in C$ and $p_5-p_1+p_6=p_2=\alpha \in C$, which gives $p_5+p_6=p_1+p_2$. In this case we get $G(t)=(p_5+p_6)t$ and $F(t)=(p_1+p_2)t=(p_5+p_6)t$ for all $t\in R$, which is conclusion (ii);

• $p_5-p_1+p_6=p_2=\alpha \in C$ and $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R which gives $p_5+p_6=p_1+p_2$ and $p_2\in C$. In this case, we get $G(t)=p_5t+tp_6=p_5t+t(p_1+p_2-p_5)=[p_5, t]+t(p_1+p_2)$, $F(t)=(p_1+p_2)t$ for all $t\in R$ and $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R, which is conclusion ((v) (a)).

Case-II: If $p_3, p_2 \in C$, then $H(t)=t(p_3+p_4)$ and $F(t)=(p_1+p_2)t$ for all $t\in R$. Then (4) reduces to

$$\begin{aligned} (p_5-p_1-p_2)\pi (\xi )^2(p_3+p_4)+\pi (\xi )^2(p_3+p_4)p_6=0, \end{aligned}$$

for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$. From Lemma 4.5, we have one of the following:

• $(p_3+p_4)p_6\in C$, $p_3+p_4\in C$ and $(p_3+p_4)p_6+(p_5-p_1-p_2)(p_3+p_4)=0$. In this case, we have $H(t)=(p_3+p_4)t$, $F(t)=(p_1+p_2)t$ for all $t\in R$. If $p_3+p_4=0$, then we get conclusion (i). If $p_3+p_4\ne 0$, then $p_6\in C$ and $p_5+p_6=p_1+p_2$. Thus we get $G(t)=(p_5+p_6)t=(p_1+p_2)t$ for all $t\in R$, which is conclusion (ii);

• $p_5-p_1-p_2\in C$ and $(p_3+p_4)p_6+(p_5-p_1-p_2)(p_3+p_4)=(p_3+p_4)(p_5+p_6-p_1-p_2)=0$. Thus we get $p_5=\lambda +p_1+p_2$ for some $\lambda \in C$. In this case, we get $H(t)=t(p_3+p_4)$, $F(t)=(p_1+p_2)t$ and $G(t)=p_5t+tp_6=\lambda t+(p_1+p_2)t+tp_6$ for all $t\in R$ with $(p_3+p_4)p_6=-(p_3+p_4)\lambda$, which is conclusion (iii);

• $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and $F(t)=(p_1+p_2)t$, $H(t)=t(p_3+p_4)$ and $G(t)=p_5t+tp_6$ with $p_5(p_3+p_4)+(p_3+p_4)p_6-(p_1+p_2)(p_3+p_4)=0$, which is conclusion ((v) (b)).

Case-III: If $p_1, p_4\in C$, then $H(t)=(p_3+p_4)t$ and $F(t)=t(p_1+p_2)$ for all $t\in R$. Then (4) reduces to

$$\begin{aligned} p_5(p_3+p_4)\pi (\xi )^2-\pi (\xi )(p_1+p_2)(p_3+p_4)\pi (\xi )+(p_3+p_4)\pi (\xi )^2p_6=0, \end{aligned}$$

for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$. By Lemma 4.6, we have one of the following:

• $p_6\in C$ and $p_5(p_3+p_4)+(p_3+p_4)p_6=(p_1+p_2)(p_3+p_4)=\alpha \in C$ for some $\alpha \in C$. In this case we have $H(t)=(p_3+p_4)t$, $F(t)=t(p_1+p_2)$ and $G(t)=(p_5+p_6)t$ for all $t\in R$ with $(p_5+p_6)(p_3+p_4)=(p_1+p_2)(p_3+p_4)=\alpha \in C$, which is conclusion (iv);

• $p_3+p_4\in C$, $p_5(p_3+p_4)\in C$ and $p_5(p_3+p_4)+(p_3+p_4)p_6=(p_1+p_2)(p_3+p_4)=\alpha \in C$ for some $\alpha \in C$. If $p_3+p_4=0$, then we get conclusion (i). Suppose $p_3+p_4\ne 0$ that is $H\ne 0$. Then $(p_5+p_6)(p_3+p_4)=(p_1+p_2)(p_3+p_4)=\alpha \in C$, which implies that $p_5+p_6=p_1+p_2$, which gives our conclusion (ii);

• $\pi (\omega _1,\ldots ,\omega _n)^2$ is central valued on R and $p_5(p_3+p_4)+(p_3+p_4)p_6=(p_1+p_2)(p_3+p_4)=\alpha \in C$ for some $\alpha \in C$, which is conclusion ((v) (b)).

Case-IV: If $p_1, p_2\in C$, then $F(t)=(p_1+p_2)t=\alpha t$. Then (4) reduces to

$$\begin{aligned}&p_5p_3\pi (\xi )^2+p_5\pi (\xi )^2p_4+p_3\pi (\xi )^2p_6+\pi (\xi )^2p_4p_6\\&-\pi (\xi )\alpha p_3 \pi (\xi )-\pi (\xi )^2\alpha p_4=0. \end{aligned}$$

From Lemma 4.7, we get $\alpha p_3\in C$.

If $\alpha \ne 0$, then $p_3\in C$. Thus we have $p_3\in C$, $p_1, p_2\in C$, which is a Case-II.

If $\alpha =0$, then $F=0$. Thus we get

$$\begin{aligned} p_5p_3\pi (\xi )^2+p_5\pi (\xi )^2p_4+p_3\pi (\xi )^2p_6+\pi (\xi )^2p_4p_6=0, \end{aligned}$$

(5)

for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$. If $\pi (\xi )^2$ is central valued on R, then $p_5(p_3+p_4)+(p_3+p_4)p_6=0$, which is conclusion ((v) (c)).

Now assume that $\pi (\xi )^2$ for all $\xi =(\omega _1,\ldots ,\omega _n)\in R^n$ is not central valued on R. Then R must be noncommutative. Suppose $R_{1}$ is a subset of R generated by $R_{2}$, where $R_{2}=\{\pi (\omega _1,\ldots ,\omega _n)^2 \mid \omega _1,\ldots ,\omega _n\in R\}$. Clearly $R_1$ is an additive subgroup of R and $R_{2}\ne \{0\}$, since $\pi (\omega _1,\ldots ,\omega _n)^2$ is noncentral valued on R. From equation (5) we get

$$\begin{aligned} p_5p_3s+p_5sp_4+p_3sp_6+sp_4p_6=0, \end{aligned}$$

(6)

for all $s\in R_{1}.$ By [10], we get one of the following

• $R_{1} \subseteq Z(R)$;

• R satisfies $s_4$ and char $(R)=2$, except when $R_{1}$ contains a noncentral Lie ideal L of R.

Since $\pi (\omega _1,\ldots ,\omega _n)^2$ is not central valued on R, the first case can not possible. Since char $(R)\ne 2$, second case also can not occur except $R_{1}$ contains a noncentral Lie ideal of R. Thus there exists Lie ideal $L\subseteq R_{1}$, where $L\not \subseteq Z(R)$. By [7, Lemma 1], there exists a noncentral two sided ideal I of R such that $[I,R]\subseteq L$. Then (6) gives that

$$\begin{aligned} p_5p_3s+p_5sp_4+p_3sp_6+sp_4p_6=0, \end{aligned}$$

for all $s\in [R, R]$. From Lemma 4.9, we get one of the following:

• R satisfies $s_4$. In this case, we get $F=0$, $G(t)=p_5t+tp_6$, $H(t)=p_3t+tp_4$ for all $t\in R$, which is conclusion (vi);

• $p_4, p_3 \in C$ and $p_5p_3+p_5p_4=-(p_4p_6+p_6p_3)\in C$. In this case, we get $p_1, p_4 \in C$, which is Case-III.

• $p_4, p_6, p_4p_6 \in C$ and $p_5p_3+p_5p_4+p_3p_6+p_4p_6=0$. In this case, we get $p_1\in C, p_4\in C$, which is Case-III.

• $p_5p_3, p_5, p_3\in C$ and $p_5p_3+p_5p_4+p_3p_6+p_4p_6=0$. In this case, we get $p_2\in C, p_3\in C$, which is Case-II.

• $p_5, p_6\in C$ and $(p_5+p_6)p_3=-(p_5+p_6)p_4 \in C$. If $p_5+p_6=0$, then $G=0$. In this case, we get $G=0$ and $F=0$, which is conclusion (vii). If $p_5+p_6\ne 0$, then $p_3, p_4 \in C$. Thus we get $p_1, p_4\in C$, which is Case-III.

• $\mu , \lambda , \eta \in C$ such that $p_6+\eta p_4=\lambda$, $p_5-\eta p_3=\mu$ and $p_5p_3+\lambda p_3=-(p_4p_6+\mu p_4) \in C$. We get $G(t)=p_5t+tp_6$, $H(t)=p_3t+tp_4$, $F=0$, which is conclusion (viii).

$\square$

5 Proof of the Theorem 2.1

In this section, throughout we shall use the following well known results.

Remark 5.1

By (Proposition 2.5.1 [3]), we can uniquely extend every derivation d to a derivation of U.

Remark 5.2

(Kharchenko [22, Theorem 2]) Suppose that R is a prime ring, $\delta \ne 0$ a derivation on R. If I is a nonzero ideal of R satisfies the differential identity

$$\begin{aligned} \pi (\omega _1,\ldots ,\omega _n,\delta (\omega _1),\ldots ,\delta (\omega _n))=0 \end{aligned}$$

for any $\omega _1,\ldots ,\omega _n \in I$, then either I satisfies the GPI

$$\begin{aligned} \pi (\omega _1,\ldots ,\omega _n,t_1,\ldots ,t_n)=0 \end{aligned}$$

or $\delta$ is U-inner i.e., for some $q \in U,\ \ \delta (t)=[q,t]$ and I satisfies the GPI

$$\begin{aligned} \pi (\omega _1,\ldots ,\omega _n,[q,\omega _1],\ldots ,[q,\omega _n])=0. \end{aligned}$$

Proof of theorem 2.1:

If $H=0$ then we are done. We assume $H\ne 0$. Then by [23, Theorem 3] there exist $q,b,a\in U$ and derivations $d,\delta ,\theta$ on U with $G(t)=qt+d(t)$, $F(t)=bt+\delta (t)$ and $H(t)=at+\theta (t)$ for all $t\in U$. Then by our hypothesis we have

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n)^2)+ad(\pi (\omega _1,\ldots ,\omega _n)^2)\nonumber \\{} & {} \quad +d\theta (\pi (\omega _1,\ldots ,\omega _n)^2)=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n)), \end{aligned}$$

(7)

which can be written as

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+d\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))d(\pi (\omega _1,\ldots ,\omega _n))+d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\nonumber \\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)d\theta (\pi (\omega _1,\ldots ,\omega _n))=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

(8)

If $d,\delta$ and $\theta$ all are inner derivations then from Proposition 4.1, the result follows. Suppose all $\delta , d$ and $\theta$ are not inner derivations together. Now we need to study the following. Case 1: Let $d(t)=[c,t]$ for all $t\in R$, where $c\in U$ i.e. d is inner. $\square$

Subcase 1a: Let $\delta$ be inner, say $\delta (x)=[p,x]$ and $\theta$ be an outer derivation. Putting these values in (8) we get

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+a[c,\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\theta (\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)[c,\theta (\pi (\omega _1,\ldots ,\omega _n)]=b\pi (\omega _1,\ldots ,x_n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

Since $\theta$ is an outer derivation on U, by replacing the value of $\theta (\pi (\omega _1,\ldots ,\omega _n))$ from (1) and by applying Remark 5.2) U satisfies

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\pi ^{\theta }(\omega _1,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +q\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)+q\pi (\omega _1,\ldots ,\omega _n)\pi ^{\theta }(\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +q\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)+a[c,\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +a\pi (\omega _1,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\pi ^{\theta }(\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +[c,\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)+\pi ^{\theta }(\omega _1,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]\\{} & {} \qquad +\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]\\{} & {} \qquad +[c,\pi (\omega _1,\ldots ,\omega _n)]\pi ^{\theta }(\omega _1,\ldots ,\omega _n))+[c,\pi (\omega _1,\ldots ,\omega _n)]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)]\\{} & {} \qquad +\pi (\omega _1,\ldots ,\omega _n)[c,\pi ^{\theta }(\omega _1,\ldots ,\omega _n)]+\pi (\omega _1,\ldots ,\omega _n)[c,\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)]\\{} & {} \quad =b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)+b\pi (\omega _1,\ldots ,\omega _n)\pi ^{\theta }(\omega _1,\ldots ,\omega _n))\\{} & {} \qquad +b\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)+[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +[p,\pi (\omega _1,\ldots ,\omega _n)]\pi ^{\theta }(\omega _1,\ldots ,\omega _n)+[p,\pi (\omega _1,\ldots ,\omega _n)]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n), \end{aligned}$$

for all $\omega _i,\eta _i\in U$. In particular U satisfies the blended component

$$\begin{aligned}{} & {} q\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +q\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]\\{} & {} \qquad +[c,\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +[c,\pi (\omega _1,\ldots ,\omega _n)]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\pi (\omega _1,\ldots ,\omega _n)[c,\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)]\\{} & {} \quad =b\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +[p,\pi (\omega _1,\ldots ,\omega _n)]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n), \end{aligned}$$

for all $\omega _i,\eta _i\in U$, where $\eta _i=\theta (\omega _i)$. By replacing $\eta _1=\omega _1$ and $\eta _i=0$ for $i=2, 3, \cdots ,n$ we get

$$\begin{aligned} 2qX^2+2X[c,X]+2[c,X]X=bX^2+[p,X]X, \end{aligned}$$

where $X=\pi (\omega _1,\ldots ,\omega _n).$ We can write it again as

$$\begin{aligned} (2q+2c)X^2-X^22c=\{(b+p)X-Xp\}X. \end{aligned}$$

Now result follows from Proposition 4.1 by taking $G(t)=(2q+2c)t-t(2c)$, $F(t)=(b+p)t-tp$ and $H(t)=t$ for all $t\in R$.

Subcase 1b: Let $\delta$ be outer and $\theta$ be inner say $\theta (t)=[s,t]$ for all $t\in R$ for some $s\in U$. Substituting these values in (7) we get

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q[s,\pi (\omega _1,\ldots ,\omega _n)^2]+a[c,\pi (\omega _1,\ldots ,\omega _n)^2]\\{} & {} \quad +[c,[s,\pi (\omega _1,\ldots ,\omega _n)^2]]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)[s,\pi (\omega _1,\ldots ,\omega _n)]+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))[s,\pi (\omega _1,\ldots ,\omega _n)]. \end{aligned}$$

Since $\delta$ is an outer derivation, by Remark 5.2, we replace $\delta (\pi (\omega _1,\ldots ,\omega _n))$ from (1) in above expression, we get

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q[s,\pi (\omega _1,\ldots ,\omega _n)^2]+a[c,\pi (\omega _1,\ldots ,\omega _n)^2]\\{} & {} \quad +[c,[s,\pi (\omega _1,\ldots ,\omega _n)^2]]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)[s,\pi (\omega _1,\ldots ,\omega _n)]+\pi ^{\delta }(\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)+\pi ^{\delta }(\omega _1,\ldots ,\omega _n)[s,\pi (\omega _1,\ldots ,\omega _n)]\\{} & {} \quad +\sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)[s,\pi (\omega _1,\ldots ,\omega _n)], \end{aligned}$$

for all $\omega _i,\sigma _i\in U$. In particular U satisfies the blended component

$$\begin{aligned} \sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)+\sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)[s,\pi (\omega _1,\ldots ,\omega _n)]. \end{aligned}$$

For $\sigma _1=\omega _1$ and $\sigma _i=0$ for $i=2, 3, \cdots , n$, we get $X^2\,s=X(a+s)X$, where $X=\pi (\omega _1,\ldots ,\omega _n)$. By taking $G(t)=ts$, $F(t)=t(a+s)$ and $H(t)=t$ for all $t\in R$, where $s\in U$ in Proposition 4.1, the result follows.

Subcase 1c: Suppose $\delta$ and $\theta$ both are outer derivations. Now two cases arise

Subcase 1c(i): The set $\{\delta ,\theta \}$ is linearly C-independent. The equation (8) is written as

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+a[c,\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\theta (\pi (\omega _1,\ldots ,\omega _n))]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)[c,\theta (\pi (\omega _1,\ldots ,\omega _n))]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

Since $\delta$ and $\theta$ are outer derivations, by Remarks 5.2, we can replace $\delta (\pi (\omega _1,\ldots ,\omega _n))$ and $\theta (\pi (\omega _1,\ldots ,\omega _n))$ from (1), where $\delta (\omega _i)=\sigma _i$, $\theta (\omega _i)=\eta _i$ in above expression, then U satisfies

$$\begin{aligned}&\sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n). \end{aligned}$$

For $\sigma _1=\eta _1=\omega _1$ and $\sigma _i=\eta _i=0$, $i\ge 2$ we have $X^2=0$, where $X=\pi (\omega _1,\ldots ,\omega _n)$ a contradiction.

Subcase 1c(ii): Suppose the set $\{\delta ,\theta \}$ is linearly C-dependent modulo inner derivation on U. Then there are $\lambda ',\mu '\in C$ and $p'\in U$ with $\lambda ' \delta (t)+\mu ' \theta (t)=[p',t]$ for all $t\in R$. In a case either $\lambda '=0$ or $\mu '=0$, we have a contradiction. Hence both $\lambda '$ and $\mu '$ can not be zero. So we can write $\delta (t)=\lambda \theta (t)+[p,t]$ where $\lambda =-{\lambda '}^{-1}\mu '$ and $p={\lambda '}^{-1}p'$. Substituting this value in (8)

$$\begin{aligned}{} & {} \left( qa+[c,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+a[c,\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\theta (\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))[c,\pi (\omega _1,\ldots ,\omega _n)]+[c,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)[c,\theta (\pi (\omega _1,\ldots ,\omega _n))]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)+\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

Since $\theta$ is an outer derivation, by application of Remark 5.2, we can replace $\theta (\pi (\omega _1,\ldots ,\omega _n))$ from (1), where $\theta (\omega _i)=\eta _i$ in above expression and then U satisfies

$$\begin{aligned}{} & {} q\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +q\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\Big [c,\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n) \Big ]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n) \Big [c, \pi (\omega _1,\ldots ,\omega _n)\Big ]\\{} & {} \qquad +\Big [c,\pi (\omega _1,\ldots ,\omega _n) \Big ]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\pi (\omega _1,\ldots ,\omega _n)\Big [c, \sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n) \Big ]\\{} & {} \quad =b\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\lambda \sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +\lambda \pi ^{\theta }(\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\lambda \sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\pi ^{\theta }(\omega _1,\ldots ,x_n)\\{} & {} \qquad +\lambda \sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n)\\{} & {} \qquad +\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]\sum \limits _i \pi (\omega _1,\ldots ,\eta _i,\ldots ,\omega _n), \end{aligned}$$

for all $\omega _i,\eta _i\in U$. For $\omega _1=0$ in above expression we get $\lambda \pi (\eta _1,\omega _2,\ldots ,\omega _n)^2=0$. Since $\lambda \ne 0$ we get $\pi (\eta _1,\omega _2,\ldots ,\omega _n)^2=0$, a contradiction.

Case 2: Let $\delta (t)=[p,t]$ for all $t\in R$ for some $p\in U$ i.e. $\delta$ be an inner derivation.

Subcase 2a: Suppose $d(t)=[c,t]$, where $c\in U$ and $\theta$ is an outer. This case is same as Subcase 1a.

Subcase 2b: Suppose $\theta (t)=[s,t]$ for all $t\in R$, $s\in U$ and d is an outer. Then expression (7) becomes

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\Big [s,\pi (\omega _1,\ldots ,\omega _n)^2\Big ]\\{} & {} \qquad +ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+\Big [d(s),\pi (\omega _1,\ldots ,\omega _n)^2\Big ]\\{} & {} \qquad +\Big [s,d(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots \omega _n)\Big ]+\Big [s,\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))\Big ]\\{} & {} \quad =b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)+b\pi (\omega _1,\ldots ,\omega _n)\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ]\\{} & {} \qquad +\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]a\pi (\omega _1,\ldots ,\omega _n)+\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ]. \end{aligned}$$

Since d is an outer, By Remark 5.2 we can replace the value of $d(\pi (\omega _1,\ldots ,\omega _n))$ from (1), where $d(\omega _i)=\nu _i$ in above expression and then U satisfies

$$\begin{aligned}&a\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\&+a\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\\&+\Big [s,\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\Big ]\\&+\Big [s,\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\Big ], \end{aligned}$$

for all $\omega _i,\nu _i\in U$. Substituting $\nu _1=\omega _1$ and $\nu _i=0$ for $i\ge 2$ in above we get $2aX^2+2[s,X^2]=0$. Then result follows from Proposition 4.1 by taking $G(t)=[2\,s,t]$, $F(t)=-2at$ and $H(t)=t$ for all $t\in R$.

Subcase 2c: Suppose d and $\theta$ both are outer derivation. Then following two cases arise.

Subcase 2c(i): The set $\{d,\theta \}$ is linearly C-independent. Substituting the value of $\delta (t)=[p, t]$ in (8) and by using Remark 5.2 we can replace $d(\pi (\omega _1,\ldots ,\omega _n))$, $\theta (\pi (\omega _1,\ldots ,\omega _n))$ and $d\theta (\pi (\omega _1,\ldots ,\omega _n))$ from (1) where $d(\omega _i)=\nu _i$, $\theta (\omega _i)=\eta _i$ and $d\theta (\omega _i)=w_i$, U satisfies the blended component

$\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)+\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n).$ By replacing $w_1=\omega _1$ and $w_i=0$ for $i=2, 3,\cdots , n$, we get $2\pi (\omega _1,\ldots ,\omega _n)^2=0$, which leads to a contradiction.

Subcase 2c(ii): The set $\{d,\theta \}$ is linearly C-dependent. Then there are $\lambda ',\mu '\in C$ and $p'\in U$ with $\lambda 'd(t)+\mu '\theta (t)=[p',t]$ for all $t\in R$. If either $\lambda '=0$ or $\mu '=0$, we get a contradiction. So consider $0\ne \lambda '$ and $0\ne \mu '$. Now we write $d(t)=-\lambda '^{-1}\mu '\theta (t)+[\lambda '^{-1}p',t]=\lambda \theta (t)+[p,t]$, where $\lambda =-\lambda '^{-1}\mu '$ and $p=\lambda '^{-1}p'$. Substituting the values of $\delta (t)=[c, t]$ and $d(t)=\lambda \theta (t)+[p,t]$ in expression (8) we get

$$\begin{aligned}{} & {} \left( qa+\lambda \theta (a)+[p,a]\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+a\lambda \theta (f(\omega _1,\ldots ,\omega _n))f(\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]\pi (\omega _1,\ldots ,\omega _n)+a\pi (\omega _1,\ldots ,\omega _n)\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)\Big [p,\pi (\omega _1,\ldots ,x_n)\Big ]+\lambda \theta ^{2}(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\Big [p,\theta (\pi (\omega _1,\ldots ,\omega _n))\Big ]\pi (\omega _1,\ldots ,\omega _n)+\theta (\pi (\omega _1,\ldots ,\omega _n))\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]+\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\Big [p,\pi (\omega _1,\ldots ,\omega _n)\Big ]\theta (\pi (\omega _1,\ldots ,\omega _n))+\pi (\omega _1,\ldots ,\omega _n)\lambda \theta ^{2}(\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)\Big [p,\theta (\pi (\omega _1,\ldots ,\omega _n))\Big ]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\Big [c,\pi (\omega _1,\ldots ,\omega _n)\Big ]a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\Big [c,\pi (\omega _1,\ldots ,\omega _n)\Big ]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

Since $\theta$ is an outer derivation, by using Remark 5.2 and then U satisfies

$$\begin{aligned}{} & {} \lambda \sum \limits _i\pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\lambda \pi (\omega _1,\ldots ,\omega _n)\sum \limits _i\pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)=0, \end{aligned}$$

for all $\omega _i,w_i\in U$, where $\theta ^2(\omega _i)=w_i$. By replacing $w_1=\omega _1$ and $w_i=0$ for $i=2, 3, \cdots , n,$ we get $2\lambda \pi (\omega _1,\ldots ,\omega _n)^2=0$ which is a contradiction.

Case 3: Suppose $\theta$ is an inner derivation, say $\theta (t)=[s,t]$ for all $t\in R$ for some $s\in U$. Then (8) gives that

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]+ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+d\Big (\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]\Big )\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]d(\pi (\omega _1,\ldots ,\omega _n))+d(\pi (\omega _1,\ldots ,\omega _n))\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]\nonumber \\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)d\Big (\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]\Big )=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ]+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\nonumber \\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))\Big [s, \pi (\omega _1,\ldots ,\omega _n)\Big ], \end{aligned}$$

(9)

for all $\omega _1,\ldots ,\omega _n\in U$. Then we have the following subcases.

Subcase 3a: Let $d(t)=[c,t]$ for all $t\in R$, where $c\in U$ and $\delta$ be an outer derivation. This case is same as the Subcase 1b.

Subcase 3b: Let d be an outer derivation and $\delta$ be an inner derivation, say $\delta (t)=[p,t]$ for all $t\in R$ for some $p\in U$. This case is same as the Subcase 2b.

Subcase 3c: Suppose d and $\delta$ both are outer derivations. Then two cases arise.

Subcase 3c(i): Let the set $\{d,\delta \}$ be linearly C-independent. Since d and $\delta$ are outer derivation, we replace $d(\pi (\omega _1,\ldots ,\omega _n))$ with and $\delta (\pi (\omega _1,\ldots ,\omega _n))$ from (1) where $d(\omega _i)=\nu _i$ and $\delta (\omega _i)=\sigma _i$ in the expression (9) and then U satisfies the blended component

$$\begin{aligned}{} & {} \sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\sum \limits _i \pi (\omega _1,\ldots ,\sigma _i,\ldots ,\omega _n)\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ]. \end{aligned}$$

By replacing $\sigma _1=\omega _1$ and $\sigma _i=0$ for $i=2, 3, \cdots , n$, we have

$$\begin{aligned} XaX+X[s,X]=0, \end{aligned}$$

where $X=\pi (\omega _1,\ldots ,\omega _n)$. Now result follows from Proposition 4.1 by taking $G(t)=0$, $F(t)=t$ and $H(t)=at+[s,t]$ for all $t\in R$.

Subcase 3c(ii): Suppose the set $\{d,\delta \}$ is linearly C-dependent. Then there are scalars $\lambda ',\mu '\in C$ and $p'\in U$ with $\lambda 'd(t)+\mu '\delta (x)=[p',t]$. In a case either $\lambda '=0$ or $\mu '=0$, we get a contradiction. Therefore we assume $0\ne \lambda '$ and $0\ne \mu '$. We can write $\delta (t)=-\mu '^{-1}\lambda 'd(t)+[\mu '^{-1}p',t]=\mu d(t)+[p,t]$ where $\mu =-\mu '^{-1}\lambda '$ and $p=-\mu '^{-1}p'$. Substituting these values in expression (9) and then applying Remark 5.2U satisfies the blended component

$$\begin{aligned}{} & {} q\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +a\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\\{} & {} \qquad +\Big [s,\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\Big ]\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ]\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\\{} & {} \qquad +\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ]\\{} & {} \qquad +\pi (\omega _1,\ldots ,\omega _n)\Big [s,\sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\Big ]\\{} & {} \quad =\mu \sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \qquad +\mu \sum \limits _i \pi (\omega _1,\ldots ,\nu _i,\ldots ,\omega _n)\Big [s,\pi (\omega _1,\ldots ,\omega _n)\Big ], \end{aligned}$$

for all $\omega _i,\nu _i\in U$, where $\nu _i=d(\omega _i)$. By replacing $\nu _1=\omega _1$ and $\nu _i=0$, $i=2, 3, \cdots , n$ the above expression becomes

$$\begin{aligned} 2aX^2+2[s,X]X+2X[s,X]=\mu XaX+\mu X[s,X], \end{aligned}$$

where $X=\pi (\omega _1,\ldots ,\omega _n)$. Last expression again can be written as $(2a+2s)X^2+X^2(\mu s-2s)=\mu X(a+s)X$. Now result follows from Proposition 4.1 by considering $G(t)=(2a+2\,s)t+t(\mu s-2\,s)$, $F(t)=t\mu (a+s)$ and $H(t)=t$ for all $t\in R$.

Case 4: We consider all derivations $d,\delta$ and $\theta$ are outer derivations. Two cases arise.

Subcase 4a: The set $\{d,\delta ,\theta \}$ is linearly C-independent. In this case by application of Remark 5.2 and using similar argument in equation (8) as we have used above, U satisfies the blended component

$$\begin{aligned}{} & {} \sum \limits _i \pi (\omega _1,\ldots ,t_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,t_i,\ldots ,\omega _n)=0, \end{aligned}$$

for all $\omega _i,t_i\in U$, where $t_i=d\theta (\omega _i)$. By replacing $t_1=\omega _1$, $t_i=0$ for $i=2, 3, \cdots , n$, we get $2\pi (\omega _1,\ldots ,\omega _n)^2=0$ which is a contradiction.

Subcase 4b: The set $\{d,\delta ,\theta \}$ is linearly dependent modulo inner derivation. So there are scalars $\lambda ',\mu ',\nu '\in C$ and $p'\in U$ such that $\lambda ' d(t)+\mu '\delta (t)+\nu '\theta (t)=[p',t]$ for all $t\in R$. If any two of $\lambda ',\mu ',\nu '$ are zero simultaneously then we get a contradiction. Thus it implies that either one of $\lambda ',\mu ',\nu '$ is zero or none of $\lambda ',\mu ',\nu '$ are zero.

Firstly, if $\lambda '=0$, $\mu '\ne 0$ and $\nu '\ne 0$ then we write $\delta (t)=\nu \theta (t)+[p,t]$ where $\nu =-\mu '^{-1}\nu '$ and $p=\mu '^{-1}p'$. Substituting these values in expression (8) we get

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+d\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))d(\pi (\omega _1,\ldots ,\omega _n))+d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)d\theta (\pi (\omega _1,\ldots ,\omega _n))=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\nu \theta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)+\nu \theta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

If d and $\theta$ are linearly C-dependent modulo inner derivation on U. By using similar argument as we have used in Subcase 2c(ii), the result follows.

If d and $\theta$ are linearly C-independent modulo inner derivation on U, then by using Remark 5.2, U satisfies

$$\begin{aligned} \pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)+\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n), \end{aligned}$$

for all $\omega _i, w_i\in U$, where $d\theta (\omega _i)=w_i$. By replacing $w_1=\omega _1$ and $w_i=0$ for $i=2, 3, \cdots , n$ we get $2\pi (\omega _1,\ldots ,\omega _n)^2=0$, which leads to a contradiction.

Secondly, if $0\ne \lambda '$, $0\ne \mu '$ and $\nu '\ne 0$ then we write $d(t)=\lambda \theta (t)+[p,t]$ for all $t\in R$, where $\lambda =-\lambda '^{-1}\nu '$ and $p=\lambda '^{-1}p'$. Substituting these values in expression (8) we get

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots \omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+a\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a[p,\pi (\omega _1,\ldots ,\omega _n)]\pi (\omega _1,\ldots ,\omega _n)+a\pi (\omega _1,\ldots ,\omega _n)\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)[p,\pi (\omega _1,\ldots ,\omega _n)]+\lambda \theta ^2(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +[p,\theta (\pi (\omega _1,\ldots ,\omega _n))]\pi (\omega _1,\ldots ,\omega _n)+\theta (\pi (\omega _1,\ldots ,\omega _n))\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))[p,\pi (\omega _1,\ldots ,\omega _n)]+\lambda \theta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,x_n))\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n))+\pi (\omega _1,\ldots ,\omega _n)\lambda \theta ^2(\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)[p,\theta (\pi (\omega _1,\ldots ,\omega _n))]=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\delta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\delta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

If $\theta$ and $\delta$ are linearly C-dependent modulo inner derivation on U. By using similar argument as we have used in Subcase 1c(i), the result follows

If d and $\delta$ are linearly C-independent modulo inner derivation on U. Then by using Remark 5.2, U satisfies

$$\begin{aligned}&\lambda \sum \limits _i\pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\&+\lambda \pi (\omega _1,\ldots ,\omega _n)\sum \limits _i\pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n), \end{aligned}$$

for all $\omega _i,w_i\in U$, where $\theta ^2(\omega _i)=w_i$. By replacing $w_1=\omega _1$ and $w_i=0$ for $i=2, 3, \cdots , n$, we get $2\lambda \pi (\omega _1,\ldots ,\omega _n)^2=0$ which is a contradiction.

Thirdly, if $\lambda '\ne 0$, $\mu '\ne 0$ and $\nu '=0$ then we write $\delta (t)=\mu d(t)+[p,t]$ for all $t\in R$, where $\mu =-\mu '^{-1}\lambda '$ and $p=\mu '^{-1}p'$. Substituting these values in expression (8) we get

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+d\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))d(\pi (\omega _1,\ldots ,\omega _n))+d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)d\theta (\pi (\omega _1,\ldots ,\omega _n))=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\mu d(\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)+\mu d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

If $\theta$ and d are linearly C-dependent modulo inner derivation on U. By using similar argument as we have used in Subcase 2c(ii), the result follows.

If d and $\theta$ are linearly C-independent modulo inner derivation on U. Then by using Remark 5.2, U satisfies

$$\begin{aligned}&\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n)\pi (\omega _1,\ldots ,\omega _n)\\&+\pi (\omega _1,\ldots ,\omega _n)\sum \limits _i \pi (\omega _1,\ldots ,w_i,\ldots ,\omega _n), \end{aligned}$$

where $d\theta (\omega _i)=w_i$. By replacing $w_1=\omega _1$ and $w_i=0$ for $i=2, 3, \cdots , n$ we get $2\pi (\omega _1,\ldots ,\omega _n)^2=0$, which is a contradiction.

Finally, consider $0\ne \lambda '$, $0\ne \mu '$ and $\nu '\ne 0$ then we write $\delta (t)=\lambda d(t)+\mu \theta (t)+[p,t]$ where $\lambda =-\mu '^{-1}\lambda '$, $\mu =-\mu '^{-1}\nu '$ and $p=\mu '^{-1}p'$. Substituting the value of $\delta (t)$ in expression (8) we get

$$\begin{aligned}{} & {} \left( qa+d(a)\right) \pi (\omega _1,\ldots ,\omega _n)^2+q\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +q\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+ad(\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +a\pi (\omega _1,\ldots ,\omega _n)d(\pi (\omega _1,\ldots ,\omega _n))+d\theta (\pi (\omega _1,\ldots ,\omega _n))\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\theta (\pi (\omega _1,\ldots ,\omega _n))d(\pi (\omega _1,\ldots ,\omega _n))+d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +\pi (\omega _1,\ldots ,\omega _n)d\theta (\pi (\omega _1,\ldots ,\omega _n))=b\pi (\omega _1,\ldots ,\omega _n)a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +b\pi (\omega _1,\ldots ,\omega _n)\theta (\pi (\omega _1,\ldots ,\omega _n))+\lambda d(\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\mu \theta (\pi (\omega _1,\ldots ,\omega _n))a\pi (\omega _1,\ldots ,\omega _n)+[p,\pi (\omega _1,\ldots ,\omega _n)]a\pi (\omega _1,\ldots ,\omega _n)\\{} & {} \quad +\lambda d(\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))+\mu \theta (\pi (\omega _1,\ldots ,\omega _n))\theta (\pi (\omega _1,\ldots ,\omega _n))\\{} & {} \quad +[p,\pi (\omega _1,\ldots ,\omega _n)]\theta (\pi (\omega _1,\ldots ,\omega _n)). \end{aligned}$$

If d and $\theta$ are linearly C-independent then it is similar to the Subcase 2c(i) and hence we get a contradiction.

If d and $\theta$ are linearly C-dependent then it is similar to the Subcase 2c(ii) and hence we get a contradiction.