Two Remarks on Generalized Skew Derivations in Prime Rings

Abstract

Let R be a prime ring of characteristic different from 2, $Q_r$ its right Martindale quotient ring, F and G two non-zero generalized skew derivations of R, associated with the same automorphism $\alpha$ and commuting with $\alpha$. In this work we describe all possible forms of F and G in the following two cases: (a) there exist $a,b \in Q_r$ and a non-central Lie ideal L of R such that $aF(x)b=0$, for all $x\in L$; (b) there exist $a_1,a_2,b_1,b_2 \in Q_r$ such that $a_1F(x)b_1+a_2G(x)b_2=0$, for all $x\in R$.

Let R be a prime ring with center Z(R), $Q_r$ its right Martindale quotient ring, C the center of $Q_r$, usually called extended centroid of R (see [1] for more details).

An   additive mapping $d:R\longrightarrow R$ is said to be a derivation of R if

$$ d(xy)=d(x)y+xd(y) $$

for all $x,y\in R$. An additive mapping $F:R\longrightarrow R$ is called a generalized derivation of R if there exists a derivation d of R such that

$$ F(xy)=F(x)y+xd(y) $$

for all $x,y\in R$.

Let R be an associative ring and $\alpha$ be an automorphism of R. An additive mapping $d:R\longrightarrow R$ is said to be a skew derivation of R if

$$ d(xy)=d(x)y+\alpha (x)d(y) $$

for all $x,y\in R$. The automorphisms $\alpha$ is called an associated automorphism of d. An additive mapping $F:R\longrightarrow R$ is called a generalized skew derivation of R if there exists a skew derivation d of R with associated automorphism $\alpha$ such that

$$ F(xy)=F(x)y+\alpha (x)d(y) $$

for all $x, y\in R$. In this case, d is called an associated skew derivation of F and $\alpha$ is called an associated automorphism of F.

In this paper we investigate some generalized differential identities involving generalized skew derivations of a prime ring of characteristic different from 2.

In [2, Theorem 2.1] Brešar describes the form of three derivations d, g, h of a prime ring R satisfying the condition $d(x)=ag(x)+h(x)b$, for any $x\in R$, where $a,b \in R \setminus Z(R)$. As a consequence he also studies the case when $ag(x)+h(x)b=0$, for any $x\in R$ [2, Corollary 2.4]. More precisely, in this last case he concludes that there exists $\lambda \in C$ such that $g(x)=[\lambda b, x]$ and $h(x)=[\lambda a, x]$, for any $x\in R$. The results by Brešar extend a theorem of Herstein contained in [12].

Following this line of investigation, J.-C. Chang generalizes the previous results to the case of both skew derivation (see [3]) and generalized skew derivations (see [4]).

Here we would like to continue the study of linear differential identities having the same flavor of the above-cited ones, and involving generalized skew derivations. In this sense, the main goal of the present paper is to prove the following theorems:

FormalPara Theorem 1

Let R be a prime ring of characteristic different from 2, F a non-zero generalized skew derivation of R, with associated automorphism $\alpha$, and a, b non-zero elements of $Q_r$ such that

$$\begin{aligned} aF(w)b=0 \quad \forall w\in L. \end{aligned}$$

Then one of the following holds:

• (a) the associated automorphism $\alpha$ is not inner and there exist $c,u \in Q_r$ be such that $F(x)=cx+\alpha (x)u$, for any $x\in R$, with $ac=ub=0$;

• (b) there exist $c,u, q\in Q_r$ and $\lambda \in C$ such that $F(x)=cx+\alpha (x)u$, for any $x\in R$, where $\alpha (x)=qxq^{-1}$, for any $x\in R$, with $a(c+\lambda q)=0$ and $(\lambda +q^{-1}u)b=0$.

FormalPara Theorem 2

Let R be a prime ring of characteristic different from 2, F, G two non-zero generalized skew derivations of R, associated with the same automorphism $\alpha$ and commuting with $\alpha$. Let $a_1,a_2,b_1,b_2$ be non-zero elements of $Q_r$ such that

$$\begin{aligned} a_1F(x)b_1+a_2G(x)b_2=0 \quad \forall x\in R. \end{aligned}$$

Then one of the following cases must occur

• (a) There exist $p,u,v,w,q \in Q_r$, where q is an invertible element, such that $F(x)=px+qxq^{-1}u$, $G(x)=vx+qxq^{-1}w$, for any $x\in R$, and one of the following holds:

  • 1. there exist $\alpha _1, \alpha _2, \alpha _3, \alpha _4 \in C$ such that $b_1=\alpha _1 b_2+\alpha _2 q^{-1}wb_2$, $q^{-1}ub_1=\alpha _3 b_2+\alpha _4 q^{-1}wb_2$ and $\alpha _1 a_1p+\alpha _3 a_1q+a_2v=\alpha _2 a_1p+\alpha _4 a_1q+a_2q=0$;

  • 2. there exist $\lambda , \alpha _1, \alpha _2, \alpha _3, \alpha _4 \in C$ such that $q^{-1}wb_2=\lambda b_2$, $b_1=(\alpha _1+\lambda \alpha _2)b_2$, $q^{-1}ub_1=(\alpha _3+\lambda \alpha _4)b_2$ and $(\alpha _1+\lambda \alpha _2)a_1p+(\alpha _3+\lambda \alpha _4)a_1q+a_2(v+\lambda q)=0$;

  • 3. there exist $0\ne \lambda \in C$ and $\beta _1, \beta _2\in C$ such that $a_1p=\lambda a_1q$, $a_2v=\beta _1 a_1q$, $a_2q=\beta _2 a_1q$ and $\lambda b_1+q^{-1}ub_1+\beta _1 b_2+\beta _2 q^{-1}wb_2=0$;

  • 4. there exist $0\ne \lambda \in C$ and $\mu , \eta \in C$ such that $a_1p=\lambda a_1q$, $a_2(v+\mu q)=\eta a_1q$, $(\lambda +q^{-1}u)b_1=-\eta b_2$ and $q^{-1}wb_2=\mu b_2$.

• (b) There exist $p,u,v,w \in Q_r$ such that $F(x)=px+\alpha (x)u$, $G(x)=vx+\alpha (x)w$, for any $x\in R$, and one of the following holds:

  • 5. $a_1p=a_2v=ub_1=wb_2=0$;

  • 6. $a_1p=a_2v=0$ and there exists $\mu \in C$ such that $ub_1=\mu wb_2$ and $a_2=-\mu a_1$;

  • 7. $ub_1=wb_2=0$ and there exists $\lambda \in C$ such that $a_1p=\lambda a_2v$ and $b_2=-\lambda b_1$;

  • 8. there exist $\lambda , \mu \in C$ such that $a_1p=\lambda a_2v$, $b_2=-\lambda b_1$, $ub_1=\mu wb_2$ and $a_2=-\mu a_1$.

• (c) There exist $p,v \in Q_r$ and $d, \delta$ skew derivations of R such that $F(x)=px+d(x)$,$G(x)=vx+\delta (x)$, for all $x\in R$, and one of the following holds:

  • 9. there exist $\vartheta \in C$ and $0\ne \eta \in C$ such that $\delta (x)=\eta d(x)$, for any $x\in R$, $a_1p=\vartheta a_2v$, $b_2=-\vartheta b_1$, and $a_1=\vartheta \eta a_2$;

  • 10. there exist $0\ne \vartheta \in C$, $0\ne \eta \in C$ and $p_0\in Q_r$ such that $\delta (x)=p_0x-\alpha (x)p_0+\eta d(x)$, for any $x\in R$, $a_1=\vartheta \eta a_2$, $b_2=-\vartheta b_1$, $p_0b_1=0$ and $\eta a_2p-a_2(v+p_0)=0$;

  • 11. there exist $\vartheta \in C$, $0\ne \eta \in C$ and $p_0, q\in Q_r$, where q is an invertible element, such that $\delta (x)=p_0x-qxq^{-1}p_0+\eta d(x)$, for any $x\in R$, $a_1=\vartheta \eta a_2$, $b_2=-\vartheta b_1$, $q^{-1}p_0b_1=\vartheta b_1$ and $\eta a_2p-a_2(v+p_0)+ \vartheta a_2q=0$.

Let us recall some basic facts which will be useful in the sequel.

FormalPara Fact 1

Let R be a prime ring, then the following statements hold:

• (a) Every generalized derivation of R can be uniquely extended to $Q_r$ [14, Theorem 3].

• (b) Any automorphism of R can be uniquely extended to $Q_r$ [7, Fact 2].

• (c) Every generalized skew derivation of R can be uniquely extended to $Q_r$ [4, Lemma 2].

FormalPara Fact 2

A generalized skew derivation having associated automorphism $\alpha$ and skew derivation d assumes the following form:

$$\begin{aligned} F(x)=ax+d(x) \end{aligned}$$

(1)

for all $x\in R$ (see [4, Lemma 2], [5, Theorem 3.1 and Corollary 3.2]).

We also need to recall some well-known results on generalized polynomial identities for prime rings involving skew derivations and automorphisms.

FormalPara Fact 3

([9]) If $\varPhi (x_i,D(x_i))$ is a generalized polynomial identity for R, where R is a prime ring and D is an outer skew derivation of R, then R also satisfies the generalized polynomial identity $\varPhi (x_i,y_i)$, where $x_i$ and $y_i$ are distinct indeterminates.

If $\varPhi (x_i,D(x_i),\alpha (x_i))$ is a generalized polynomial identity for a prime ring R, D is an outer skew derivation of R and $\alpha$ is an outer automorphism of R, then R also satisfies the generalized polynomial identity $\varPhi (x_i,y_i,z_i)$, where $x_i$, $y_i$, and $z_i$ are distinct indeterminates.

FormalPara Fact 4

([13, Theorem 6.5.9, page 365]) Let a prime ring R obey a polynomial identity of the type $f( x_j^{\alpha _i \Delta _k}) = 0$, where $f( z_j^{i,k})$ is a generalized polynomial with the coefficients from $Q_r$, $\Delta _1,\ldots ,\Delta _n$ are mutually different correct words from a reduced set of skew derivations commuting with all the corresponding automorphisms, and $\alpha _1,\ldots ,\alpha _m$ are mutually outer automorphisms. In this case the identity $f( z_j^{i,k})=0$ is valid on $Q_r$.

FormalPara Fact 5

([8, Theorem 1]) Let R be a prime ring and I be a two-sided ideal of R. Then I, R, and $Q_r$ satisfy the same generalized polynomial identities with coefficients in $Q_r$ (see [6]). Furthermore, I, R, and $Q_r$ satisfy the same generalized polynomial identities with automorphisms.

FormalPara Fact 6

([9, Theorem 2]) Let R be a prime ring and I be a two-sided ideal of R. Then I, R, and $Q_r$ satisfy the same generalized polynomial identities with a single skew derivation.

In the sequel, R will be a non-commutative ring of characteristic different from 2, F and G two non-zero generalized skew derivations of R, associated with the same automorphism $\alpha$ and commuting with $\alpha$.

1 Annihilating Condition for a Single Generalized Skew Derivation

In this second section our aim will be to prove Theorem 1. More precisely, let F be a generalized skew derivation of R and a, b are non-zero elements of R such that

$$\begin{aligned} aF(w)b=0 \quad \forall w\in L\quad \text{ a } \text{ non-central } \text{ Lie } \text{ ideal } \text{ of }\quad R. \end{aligned}$$

(2)

The study of this result will be useful for the proof of our main Theorem (i.e., Theorem 2).

We permit the following:

Lemma 1

Let R be a prime and $a_i, b_i \in U$, for $1\le i \le n$. If $\sum _{i=1}^n a_i[x,y]b_i=0$, for all $x,y\in R$. If $a_i\ne 0$ for some i, then $b_1,\ldots ,b_n$ are C-dependent. Similarly, if $b_i\ne 0$ for some i, then $a_1,\ldots ,a_n$ are C-dependent.

Proof

The result follows easily from [15, Lemma 2.2] and [16, Lemma 1].

Lemma 2

Let $c,u \in Q_r$ be such that $F(x)=cx+\alpha (x)u$, for any $x\in R$. If

$$\begin{aligned} aF([r_1,r_2])b=0 \quad \forall r_1,r_2\in R. \end{aligned}$$

(3)

then one of the following holds:

• (a) $ac=ub=0$;

• (b) there exist $q\in Q_r$ and $\lambda \in C$ such that $\alpha (x)=qxq^{-1}$, for any $x\in R$, with $a(c+\lambda q)=0$ and $(\lambda +q^{-1}u)b=0$.

Proof

By our assumption R satisfies

$$\begin{aligned} a\biggl (c[x_1,x_2]+\alpha ([x_1,x_2])u\biggr )b. \end{aligned}$$

(4)

We consider firstly the case $\alpha (x)=qxq^{-1}$, for any $x\in R$, where $q\in Q_r$ is an invertible element. In this case, by (4), R satisfies

$$\begin{aligned} a\biggl (c[x_1,x_2]+q[x_1,x_2]q^{-1}u\biggr )b. \end{aligned}$$

(5)

A direct application of Lemma 1 leads to conclusion (b).

Therefore we may assume that $\alpha$ is not an inner automorphism of $Q_r$. Thus, by (4) and Fact 3, R satisfies the generalized polynomial identity

$$\begin{aligned} a\biggl (c[x_1,x_2]+[y_1,y_2]u\biggr )b. \end{aligned}$$

(6)

In particular R satisfies both the blended components $ac[x_1,x_2]b$ and $a[y_1,y_2]ub$. Since $a\ne 0$ and $b\ne 0$ and by the primeness of R, we get the required conclusion $ac=ub=0$.

Proof

(Proof of Theorem 1) By Fact 2, $F(x)=cx+d(x)$ for all $x\in R$, where $c\in Q_r$ and d is the skew derivation associated with F.

Since L is not central and $char(R)\ne 2$, it is well known that there exists a non-zero ideal I of R such that $0\ne [I,R]\subseteq L$ (see [11, pages 4–5]). Therefore, by (2), the ideal I satisfies $aF([x_1,x_2])b$. Since R and I satisfy the same generalized identities with automorphisms and skew derivations, we may assume that R also satisfies $aF([x_1,x_2])b$, that is

$$\begin{aligned} a\biggl (c[x_1,x_2]+d([x_1,x_2])\biggr )b. \end{aligned}$$

(7)

In case d is an inner skew derivation of R, the conclusion follows from Lemma 2. Then we may assume that d is not inner and prove that a contradiction follows. Expansion of (7) says that R satisfies

$$\begin{aligned} a\biggl (c[x_1,x_2]+d(x_1)x_2+\alpha (x_1)d(x_2)-d(x_2)x_1-\alpha (x_2)d(x_1)\biggr )b. \end{aligned}$$

(8)

Since d is not inner and by Fact 3, (8) implies that R satisfies

$$\begin{aligned} a\biggl (c[x_1,x_2]+y_1x_2+\alpha (x_1)y_2-y_2x_1-\alpha (x_2)y_1\biggr )b \end{aligned}$$

(9)

and in particular R satisfies

$$\begin{aligned} a\biggl (y_1x_2-\alpha (x_2)y_1\biggr )b. \end{aligned}$$

(10)

If $\alpha$ is outer, relation (10) implies that R satisfies

$$\begin{aligned} a\biggl (y_1x_2-z_2y_1\biggr )b \end{aligned}$$

and, in particular, $a[r_1,r_2]b=0$, for any $r_1, r_2 \in R$. It follows that either $a=0$ or $b=0$, which contradicts the assumption $a,b\ne 0$.

On the other hand, if $\alpha (x)=qxq^{-1}$, where q is an invertible element of $Q_r$, one may replace in (main-8) $y_1$ with $qx_1$. Hence R satisfies $aq[x_1,x_2]b$. Since q is invertible, once again the contradiction that either $a=0$ or $b=0$ follows.

2 Annihilating Conditions for Two Generalized Skew Derivations

We conclude our paper giving the description of two generalized skew derivations F and G of a prime ring R satisfying the condition

$$\begin{aligned} a_1F(x)b_1+a_2G(x)b_2=0 \quad \forall x\in R \end{aligned}$$

(11)

where $a_1,a_2,b_1,b_2\in Q_r$.

In light of Theorem 1, we may assume that $a_1,a_2,b_1,b_2$ are all non-zero elements of $Q_r$ and also that both $F\ne 0$ and $G\ne 0$.

We start with two useful results, that we quote as follows, by applying [6, Theorem 2]:

Lemma 3

Let R be a prime and $a_i, b_i \in Q_r$, for $1\le i \le n$. If $\sum _{i=1}^n a_ixb_i=0$, for all $x\in R$, and $b_i\ne 0$ for some i, then $a_1,\ldots ,a_n$ are C-dependent (see [15, Lemma 2.2]).

Lemma 4

Let R be a prime and $a_i, b_i, c_i, d_i \in Q_r$ such that $\sum _{i=1}^m a_ixb_i+\sum _{j=1}^n c_jxd_j=0$, for all $x\in R$. If $a_1,\ldots ,a_m$ are linearly C-independent then each $b_i$ is a linear combination of $d_1,\ldots ,d_n$ over C. Analogously, if $b_1,\ldots ,b_m$ are linearly C-independent then each $a_i$ is a linear combination of $c_1,\ldots ,c_n$ over C. (see [17, Lemma 1.2]).

Lemma 5

Let F and G be inner generalized skew derivations of R defined as

$$\begin{aligned} F(x)=px+qxq^{-1}u, \quad G(x)=vx+qxq^{-1}w, \quad \forall x\in R \end{aligned}$$

where $p,u,v,w,q \in Q_r$ and q is an invertible element. If R satisfies (11), one of the following holds:

• (a) there exist $\alpha _1, \alpha _2, \alpha _3, \alpha _4 \in C$ such that $b_1=\alpha _1 b_2+\alpha _2 q^{-1}wb_2$, $q^{-1}ub_1=\alpha _3 b_2+\alpha _4 q^{-1}wb_2$ and $\alpha _1 a_1p+\alpha _3 a_1q+a_2v=\alpha _2 a_1p+\alpha _4 a_1q+a_2q=0$;

• (b) there exist $\lambda , \alpha _1, \alpha _2, \alpha _3, \alpha _4 \in C$ such that $q^{-1}wb_2=\lambda b_2$, $b_1=(\alpha _1+\lambda \alpha _2)b_2$, $q^{-1}ub_1=(\alpha _3+\lambda \alpha _4)b_2$ and $(\alpha _1+\lambda \alpha _2)a_1p+(\alpha _3+\lambda \alpha _4)a_1q+a_2(v+\lambda q)=0$;

• (c) there exist $0\ne \lambda \in C$ and $\beta _1, \beta _2\in C$ such that $a_1p=\lambda a_1q$, $a_2v=\beta _1 a_1q$, $a_2q=\beta _2 a_1q$ and $\lambda b_1+q^{-1}ub_1+\beta _1 b_2+\beta _2 q^{-1}wb_2=0$;

• (d) there exist $0\ne \lambda \in C$ and $\mu , \eta \in C$ such that $a_1p=\lambda a_1q$, $a_2(v+\mu q)=\eta a_1q$, $(\lambda +q^{-1}u)b_1=-\eta b_2$ and $q^{-1}wb_2=\mu b_2$.

Proof

By our main hypothesis

$$\begin{aligned} a_1F(x)b_1+a_2G(x)b_2=0 \quad \forall x\in R. \end{aligned}$$

Under the assumptions of the present Lemma, we have that R satisfies the generalized identity

$$\begin{aligned} a_1\bigl (px+qxq^{-1}u\bigr )b_1+a_2\bigl (vx+qxq^{-1}w\bigr )b_2 \end{aligned}$$

(12)

that is

$$\begin{aligned} (a_1p)xb_1+(a_1q)x(q^{-1}ub_1)+(a_2v)xb_2+(a_2q)x(q^{-1}wb_2). \end{aligned}$$

(13)

By Lemma 3 and since $a_1,a_2,b_1,b_2$ are all non-zero we may divide the proof in two cases.

Case 1. $\{a_1p, a_1q\}$ is a linearly C-independent set

Application of Lemma 4 implies that there exist $\alpha _1, \alpha _2, \alpha _3, \alpha _4 \in C$ such that

$$\begin{aligned} \begin{aligned}&b_1=\alpha _1 b_2+\alpha _2 q^{-1}wb_2\\&q^{-1}ub_1=\alpha _3 b_2+\alpha _4 q^{-1}wb_2. \end{aligned} \end{aligned}$$

(14)

Thus, by (13), R satisfies

$$\begin{aligned} (a_1p)x(\alpha _1 b_2+\alpha _2 q^{-1}wb_2)+(a_1q)x(\alpha _3 b_2+\alpha _4 q^{-1}wb_2)+(a_2v)xb_2+(a_2q)x(q^{-1}wb_2) \end{aligned}$$

that is

$$\begin{aligned} (\alpha _1 a_1p+\alpha _3 a_1q+a_2v)xb_2+(\alpha _2 a_1p+\alpha _4 a_1q+a_2q)x q^{-1}wb_2. \end{aligned}$$

(15)

Firstly we note that, if $\alpha _2 a_1p+\alpha _4 a_1q+a_2q=0$ then, by the primeness of R and since $b_2\ne 0$, (15) implies $\alpha _1 a_1p+\alpha _3 a_1q+a_2v=0$. Hence, in consideration of what is stated in relations (14), we get conclusion (a) of the present Lemma.

On the other hand, if $\alpha _2 a_1p+\alpha _4 a_1q+a_2q\ne 0$ and by Lemma 3, there is $\lambda \in C$ such that $q^{-1}wb_2=\lambda b_2$. Thus (15) reduces to

$$\begin{aligned} (\alpha _1 a_1p+\alpha _3 a_1q+a_2v)xb_2+\lambda (\alpha _2 a_1p+\alpha _4 a_1q+a_2q)x b_2. \end{aligned}$$

(16)

Again by the primeness of R and since $b_2\ne 0$, $\alpha _1 a_1p+\alpha _3 a_1q+a_2v+\lambda (\alpha _2 a_1p+\alpha _4 a_1q+a_2q)=0$ follows.

Case 2. $a_1p=\lambda a_1q$, $0\ne \lambda \in C$

In this case, again by (13), R satisfies

$$\begin{aligned} a_1qx(\lambda b_1+q^{-1}ub_1)+(a_2v)xb_2+(a_2q)x(q^{-1}wb_2). \end{aligned}$$

(17)

Notice that, in case $\{b_2, q^{-1}wb_2\}$ is a linearly C-independent set, by (17) and Lemma 3, it follows

$$a_2v=\beta _1 a_1q,\quad a_2q=\beta _2 a_1q \quad \beta _1, \beta _2 \in C$$

and (17) reduces to

$$a_1qx(\lambda b_1+q^{-1}ub_1+\beta _1 b_2+\beta _2 q^{-1}wb_2).$$

Therefore, since $a_1q\ne 0$, we get $\lambda b_1+q^{-1}ub_1+\beta _1 b_2+\beta _2 q^{-1}wb_2=0$.

Assume finally that $\{b_2, q^{-1}wb_2\}$ is a linearly C-dependent set.

Without loss of generality we may write $q^{-1}wb_2=\mu b_2$, for a suitable $\mu \in C$. Hence, by (17), R satisfies

$$\begin{aligned} a_1qx(\lambda b_1+q^{-1}ub_1)+(a_2v+\mu a_2q)xb_2 \end{aligned}$$

(18)

implying that there exists $\eta \in C$ such that

$$a_2v+\mu a_2q=\eta a_1q$$

$$\lambda b_1+q^{-1}ub_1=-\eta b_2.$$

Lemma 6

Let F and G be inner generalized skew derivations of R defined as

$$\begin{aligned} F(x)=px+\alpha (x)u, \quad G(x)=vx+\alpha (x)w, \quad \forall x\in R \end{aligned}$$

where $p,u,v,w \in Q_r$ and $\alpha$ is an outer automorphism of R. If R satisfies (11), one of the following holds:

• (a) $a_1p=a_2v=ub_1=wb_2=0$;

• (b) $a_1p=a_2v=0$ and there exists $\mu \in C$ such that $ub_1=\mu wb_2$ and $a_2=-\mu a_1$;

• (c) $ub_1=wb_2=0$ and there exists $\lambda \in C$ such that $a_1p=\lambda a_2v$ and $b_2=-\lambda b_1$;

• (d) there exist $\lambda , \mu \in C$ such that $a_1p=\lambda a_2v$, $b_2=-\lambda b_1$, $ub_1=\mu wb_2$ and $a_2=-\mu a_1$.

Proof

Here R satisfies

$$\begin{aligned} a_1\bigl (px+\alpha (x)u\bigr )b_1+a_2\bigl (vx+\alpha (x)w\bigr )b_2. \end{aligned}$$

(19)

Since $\alpha$ is outer, by (19), it follows that R satisfies the generalized identity

$$\begin{aligned} a_1\bigl (px_1+x_2u\bigr )b_1+a_2\bigl (vx_1+x_2w\bigr )b_2. \end{aligned}$$

(20)

In particular, both

$$\begin{aligned} a_1px_1b_1+a_2vx_1b_2 \end{aligned}$$

(21)

and

$$\begin{aligned} a_1x_2ub_1+a_2x_2wb_2 \end{aligned}$$

(22)

are satisfied by R. Relation (21) implies that

• either $a_1p=a_2v=0$

• or there exists $\lambda \in C$ such that $a_1p=\lambda a_2v$ and $b_2=-\lambda b_1$.

Analogously, (22) implies that

• either $ub_1=wb_2=0$

• or there exists $\mu \in C$ such that $ub_1=\mu wb_2$ and $a_2=-\mu a_1$.

Putting together all the previous informations, one of the following cases must occur:

• (a) $a_1p=a_2v=ub_1=wb_2=0$;

• (b) $a_1p=a_2v=0$ and there exists $\mu \in C$ such that $ub_1=\mu wb_2$ and $a_2=-\mu a_1$;

• (c) $ub_1=wb_2=0$ and there exists $\lambda \in C$ such that $a_1p=\lambda a_2v$ and $b_2=-\lambda b_1$;

• (d) there exist $\lambda , \mu \in C$ such that $a_1p=\lambda a_2v$, $b_2=-\lambda b_1$, $ub_1=\mu wb_2$ and $a_2=-\mu a_1$.

Before proceeding with the proof of our main result, we need to recall the following:

Lemma 7

Let R be a prime ring, $\alpha , \beta \in \textrm{Aut}(Q_r)$ and $d, \delta :R\rightarrow R$ be two skew derivations, associated with the same automorphism $\alpha$. If there exist $0\ne \eta \in C$, and $u\in Q_r$ such that

$$\begin{aligned} \delta (x)=\biggl (ux-\beta (x)u\biggr )+\eta d(x),\quad \forall x\in R \end{aligned}$$

(23)

then either $\alpha =\beta$ or $\delta (x)=\eta d(x)$, for all $x\in R$.

Proof

By the definition of $\delta$ we have

$$\begin{aligned} \delta (xy) = uxy-\beta (x)\beta (y)u+\eta d(x)y+\eta \alpha (x)d(y). \end{aligned}$$

(24)

On the other hand, right multiplying relation (23) by $y\in R$, it follows that

$$\begin{aligned} \delta (x)y = uxy-\beta (x)uy+\eta d(x)y\quad \forall x,y\in R. \end{aligned}$$

(25)

Therefore, subtracting relation (25) from (24), and using again (23), we get

$$\begin{aligned} \bigl \{\alpha (x)-\beta (x)\bigr \}\cdot \bigl \{uy-\beta (y)u\bigr \}=0\quad \forall x,y\in R. \end{aligned}$$

(26)

Replacing y by yt in (26) and then using (26) we have

$$\begin{aligned} \bigl \{\alpha (x)-\beta (x)\bigr \}\cdot \beta (y)\cdot \bigl \{\beta (t)u-ut\bigr \}=0\quad \forall x,y,t\in R. \end{aligned}$$

(27)

Then, by the primeness of R, above relation yields either $\alpha (x)-\beta (x)=0$ for any $x\in R$, or $\beta (t)u-ut=0$ for any $t\in R$. The last case and (23) imply $\delta (x)=\eta d(x)$, for all $x\in R$, as required.

Lemma 8

([10, Lemma 3.2]) Let R be a prime ring, $\alpha , \beta \in \textrm{Aut}(Q_r)$ and $d:R\rightarrow R$ be a skew derivation, associated with the automorphism $\alpha$. If there exist $0\ne \theta \in C$, $0\ne \eta \in C$ and $u,b\in Q_r$ such that

$$\begin{aligned} d(x)=\theta \biggl (ux-\alpha (x)u\biggr )+\eta \biggl (bx-\beta (x)b\biggr ),\quad \forall x\in R \end{aligned}$$

then d is an inner skew derivation of R. More precisely, either $b=0$ or $\alpha =\beta$.

Proof

(Proof of Theorem 2) For sake of clearness we recall that we may write $F(x)=px+d(x)$ and $G(x)=vx+\delta (x)$, for all $x\in R$ and suitable $p,v \in Q_r$ and $d,\delta$ skew derivations associated with the same automorphism $\alpha$. Moreover we also recall that both d and $\delta$ commute with $\alpha$.

We also remind that, by our main hypothesis R satisfies

$$\begin{aligned} a_1\biggl (px+d(x)\biggr )b_1+a_2\biggl (vx+\delta (x)\biggr )b_2. \end{aligned}$$

(28)

The case $d=0$ and $\delta \ne 0$

We firstly study the case $F(x)=px$ and $G(x)=vx+\delta (x)$, for all $x\in R$. Since $F\ne 0$, we may assume in what follows $p\ne 0$. Moreover $\delta$ is not an inner skew derivation of R, otherwise the conclusion follows by Lemmas 5 and 6. In this situation, by (28) we have that R satisfies

$$\begin{aligned} a_1px_1b_1+a_2\biggl (vx_1+x_2\biggr )b_2. \end{aligned}$$

In particular $a_2yb_2=0$, for any $y\in R$, which is a contradiction, since both $a_2\ne 0$ and $b_2\ne 0$.

Analogously, we get a contradiction in the case we assume $\delta =0$ and $d\ne 0$.

The case $d\ne 0$, $\delta \ne 0$

Here we study the case when $F(x)=px+d(x)$ and $G(x)=vx+\delta (x)$, for all $x\in R$. We start with the case $d, \delta$ are linearly C-independent modulo inner skew derivations. Hence, by (28),

$$\begin{aligned} a_1\biggl (px_1+x_2\biggr )b_1+a_2\biggl (vx_1+x_3\biggr )b_2 \end{aligned}$$

(29)

is satisfied by R. In particular, $a_1x_2b_1$ is a generalized identity for R, which is a contradiction, since both $a_1\ne 0$ and $b_1\ne 0$.

Thus we assume that $\{d,\delta \}$ are linearly C-dependent modulo inner skew derivations. Hence there exist $\lambda , \mu \in C$, $u\in Q_r$ and an automorphism $\beta$ of R such that $\lambda d(x)+\mu \delta (x)=ux-\beta (x)u$, for any $x \in R$.

If $\lambda =0$ and $\mu \ne 0$, we write

$$\begin{aligned} \delta (x)=\biggl (p_0x-\beta (x)p_0\biggr ),\quad \forall x\in R \end{aligned}$$

where $p_0=\mu ^{-1}u$. Since the automorphism associated with a skew derivation is unique, in this case $\alpha =\beta$.

If d is also inner, the conclusion follows from Lemmas 5 and 6. Hence we may assume that d is not inner. Thus, by (28), R satisfies

$$\begin{aligned} a_1\biggl (px_1+x_2\biggr )b_1+a_2\biggl (vx_1+p_0x_1-\beta (x_1)p_0\biggr )b_2 \end{aligned}$$

(30)

and in particular $a_1x_2b_1$ is an identity for R, which is a contradiction.

Similarly, we get a contradiction in the case $\mu =0$ and $\lambda \ne 0$.

Hence, in the sequel we assume that both $\lambda \ne 0$ and $\mu \ne 0$. We may write

$$\begin{aligned} \delta (x)=\biggl (p_0x-\beta (x)p_0\biggr )+\eta d(x),\quad \forall x\in R \end{aligned}$$

(31)

where $\eta =-\lambda \mu ^{-1}\ne 0$ and, as above, $p_0=\mu ^{-1}u$. By Lemma 7, either $\alpha =\beta$ or $p_0=0$ and $\delta (x)=\eta d(x)$, for all $x\in R$.

Moreover, by Lemma 8, if d is an inner skew derivation, then also $\delta$ is inner and the conclusion follows again from Lemmas 5 and 6.

Therefore, in what follows we assume that $0\ne d$ is outer.

In the case $\delta =\eta d$, (28) reduces to

$$\begin{aligned} a_1\biggl (px+d(x)\biggr )b_1+a_2\biggl (vx+\eta d(x)\biggr )b_2. \end{aligned}$$

(32)

Thus, since d is not inner, R satisfies

$$\begin{aligned} a_1\biggl (px_1+x_2\biggr )b_1+a_2\biggl (vx_1+\eta x_2\biggr )b_2. \end{aligned}$$

(33)

In particular, both

$$\begin{aligned} a_1px_1b_1+a_2vx_1b_2 \end{aligned}$$

(34)

and

$$\begin{aligned} a_1x_2b_1+\eta a_2x_2b_2 \end{aligned}$$

(35)

are identities for R. Those relations imply that there exists $\vartheta \in C$ such that

$$a_1p=\vartheta a_2v \quad b_2=-\vartheta b_1 \quad a_1=\vartheta \eta a_2.$$

Suppose now $\alpha =\beta$. By relations (31) and (28) R satisfies

$$\begin{aligned} a_1\biggl (px+d(x)\biggr )b_1+a_2\biggl (vx+p_0x-\alpha (x)p_0+\eta d(x)\biggr )b_2. \end{aligned}$$

(36)

Since d is not inner, it follows that

$$\begin{aligned} a_1\biggl (px_1+x_2\biggr )b_1+a_2\biggl (vx_1+p_0x_1-\alpha (x_1)p_0+\eta x_2\biggr )b_2 \end{aligned}$$

(37)

is a generalized identity for R. Hence R satisfies both

$$\begin{aligned} a_1px_1b_1+a_2\biggl (vx_1+p_0x_1-\alpha (x_1)p_0\biggr )b_2 \end{aligned}$$

(38)

and

$$\begin{aligned} a_1x_2b_1+\eta a_2x_2b_2. \end{aligned}$$

(39)

By (39) and applying Lemma 3, we have that there exists $0\ne \vartheta \in C$ such that

$$a_1=\vartheta \eta a_2\quad b_2=-\vartheta b_1.$$

Substituting $a_1$ and $b_2$ in relation (38), it follows that

$$\begin{aligned} \vartheta \eta a_2px_1b_1-\vartheta a_2\biggl (vx_1+p_0x_1-\alpha (x_1)p_0\biggr )b_1. \end{aligned}$$

(40)

If $\alpha$ is not inner, by (40) we have that R satisfies

$$\begin{aligned} \vartheta \eta a_2px_1b_1-\vartheta a_2\biggl (vx_1+p_0x_1-x_2p_0\biggr )b_1. \end{aligned}$$

(41)

Thus both $a_2x_2p_0b_1$ and

$$\begin{aligned} \biggl (\vartheta \eta a_2p-\vartheta a_2(v+p_0)\biggr )x_1b_1 \end{aligned}$$

are identities for R, implying $p_0b_1=0$ and $\eta a_2p-a_2(v+p_0)=0$.

On the other hand, if $\alpha (x)=qxq^{-1}$, for any $x\in R$, by (40) it follows that

$$\begin{aligned} \biggl (\eta a_2p-a_2(v+p_0)\biggr )x_1b_1 + a_2qx_1q^{-1}p_0b_1 \end{aligned}$$

is a generalized identity for R. Thus, there exists $\vartheta \in C$ such that

$$q^{-1}p_0b_1=\vartheta b_1 \quad \eta a_2p-a_2(v+p_0)+ \vartheta a_2q=0.$$