On Functional Equation Related to a Class of Generalized Inner Derivations in Prime Rings

Abstract

The main purpose of this paper is to prove the following result. Let R be a prime ring of $\mathrm{char}(R)=0$ or $\mathrm{char}(R)>4,$ and let $T{\text {:}}\,R\rightarrow R$ be an additive mapping satisfying the relation $T(x^{4})=T(x)x^{3}-xT(x^{2})x+x^{3}T(x)$ for all $x\in R.$ In this case, T is of the form $4T(x)=qx+xq$ for all $x\in R,$ where q is some fixed element from the symmetric Martindale ring of quotients of R. Some functional equations related to bicircular projections are also investigated.

This paper is a continuation of our recent work [1]. Throughout, R will represent an associative ring with center Z(R). Given an integer $n>1,$ a ring R is said to be n-torsion free, if for $x\in R,\,nx=0$ implies $x=0.$ As usual, the commutator $xy-yx$ will be denoted by $[ x,\,y].$ An additive mapping $x\mapsto x^{*}$ on a ring R is called involution in case $(xy)^{*}=y^{*}x^{*}$ and $x^{**}=x$ hold for all $x,\,y\in R.$ A ring equipped with an involution is called a ring with involution or $^{*}$-ring. Recall that a ring R is prime if for $a,\,b\in R,\,aRb=(0)$ implies that either $a=0$ or $b=0,$ and is semiprime in case $aRa=(0)$ implies $a=0.$ We denote by $\mathrm{char}(R)$ the characteristic of a prime ring R. We denote by $Q_\mathrm{mr},\,Q_\mathrm{s},$ and C the maximal Martindale right ring of quotients, symmetric Martindale ring of quotients, and extended centroid of a semiprime ring R,  respectively (see [2]). In case R is a semiprime $^{*}$-ring, then the involution can be uniquely extended to $Q_\mathrm{s}$ (see Proposition 2.5. 4 in [2]). Given some $X\subset R$, we denote $C(X)=\{ r\in R;\,[r,\,X] =0\}.$ An additive mapping $D{\text {:}}\,R\rightarrow R$ is called a derivation if $D(xy)=D(x)y+xD(y)$ holds for all pairs $x,\,y\in R,$ and is called a Jordan derivation in case $D(x^{2})=D(x)x+xD(x)$ is fulfilled for all $x\in R.$ A derivation $D{\text {:}}\,R\rightarrow R$ is inner in case D is of the form $D(x)=[a,\,x]$ for all $x\in R$ and some fixed $a\in R.$ Every derivation is Jordan derivation. The converse is in general not true. A classical result of Herstein [3] asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein theorem can be found in [4]. Cusack [5] generalized Herstein theorem to 2-torsion-free semiprime rings (see [6] for an alternative proof). It should be mentioned that Herstein theorem has been fairly generalized by Beidar et al. [7]. Generalizations of Herstein theorem can be found in [8, 9]. A mapping $T(x)=ax+xb,$ where a and b are fixed elements of a ring, is sometimes called a generalized derivation. We follow Hvala [10] and call such mappings generalized inner derivations, as they present a generalization of the concept of inner derivation. In the theory of operator algebras, they are considered as an important class of so-called elementary operators-i.e., mappings of the form

$$\begin{aligned} x\longmapsto \sum \limits _{i=1}^{n}a_{i}xb_{i}. \end{aligned}$$

We refer the reader to [11] for a good account of the theory of elementary operators. A mapping $T(x)=ax+xa,$ where a is a fixed element of a ring, will be called symmetric generalized inner derivation. Let X be a complex Banach space, and let $\mathcal {L}(X)$ denote the algebra of all bounded linear operators on X. A projection $P\in \mathcal {L} (X)$ is bicircular in case all mappings of the form $\mathrm{e}^{i\alpha }P+\mathrm{e}^{i\beta }(I-P),$ where I denotes the identity operator, are isometric for all pairs of real numbers $\alpha ,\,\beta .$

Let us start with the following result proved by Brešar [12] (see [13] for a generalization).

FormalPara Theorem 1

Let R be a 2-torsion-free semiprime ring and let $D{\text {:}}\,R\rightarrow R$ be an additive mapping satisfying the relation

$$\begin{aligned} D(xyx)=D(x)yx+xD(y)x+xyD(x), \end{aligned}$$

(1)

for all pairs $x,\,y\in R.$ In this case, D is a derivation.

Note that in case, a ring has the identity element, the proof of the result above is immediate. Namely, in this case, the substitution $y=e$ in the relation (1), where e stands for the identity element, gives that D is a Jordan derivation, and then it follows from Cusack’s generalization of Herstein theorem, that D is a derivation. An additive mapping satisfying the relation (1) on arbitrary ring is called a Jordan triple derivation. It is easy to prove that any Jordan derivation on a 2-torsion-free ring is a Jordan triple derivation (see, for example, [4] for the details), which means that Theorem 1 generalizes Cusack’s generalization of Herstein theorem.

Motivated by Theorem 1, Vukman et al. [14] have proved the following result (see [15] for a generalization).

FormalPara Theorem 2

Let R be a 2-torsion-free semiprime ring and let $T{\text {:}}\,R\rightarrow R$ be an additive mapping satisfying the relation

$$\begin{aligned} T(xyx)=T(x)yx-xT(y)x+xyT(x), \end{aligned}$$

(2)

for all pairs $x,\,y\in R.$ In this case, T is of the form $2T(x)=qx+xq$ for all $x\in R$ and some fixed $q\in Q_\mathrm{s}.$

Since any symmetric generalized inner derivation satisfies the functional equation (2), Theorem 2 characterizes symmetric generalized inner derivations among all additive mappings on 2-torsion-free semiprime rings. The substitution $y=x$ in (1) and (2) gives

$$\begin{aligned} D\left( x^{3}\right) =D(x)x^{2}+xD(x)x+x^{2}D(x), \end{aligned}$$

(3)

and

$$\begin{aligned} T\left( x^{3}\right) =T(x)x^{2}-xT(x)x+x^{2}T(x). \end{aligned}$$

(4)

The functional equation (3) has been considered in [7, 16] (actually, in [16] much more general situation has been studied). The functional equation (4) has been investigated in [17] (see [18] for a generalization). It is our aim in this paper to prove the following result.

FormalPara Theorem 3

Let R be a prime ring with $\mathrm{char}(R)=0$ or $\mathrm{char}(R)>4$ and let $T{\text {:}}\,R\rightarrow R$ be an additive mapping satisfying the relation

$$\begin{aligned} T\left( x^{4}\right) =T(x)x^{3}-xT\left( x^{2}\right) x+x^{3}T(x), \end{aligned}$$

(5)

for all $x\in R.$ In this case, T is of the form $4T(x)=qx+xq$ for all $x\in R$ and some fixed $q\in Q_\mathrm{s}.$

Obviously, the functional equation (5) is obtained by putting $x^{2}$ instead of y in (2). The functional equation $D(x^{4})=D(x)x^{3}+xD(x^{2})x+x^{3}D(x)$ obtained in the same way from (1) has been considered in our recent paper [1].

In the proof of Theorem 3, we use as the main tool the theory of functional identities (Beidar–Brešar–Chebotar theory). The theory of functional identities considers set-theoretic maps on rings that satisfy some identical relations. When treating such relations, one usually concludes that the form of the mappings involved can be described, unless the ring is very special. We refer the reader to [19] for an introductory account on functional identities and to [20] for a full treatment of this theory. For the proof of Theorem 3, we need Theorem 4 which might be of independent interest. Let R be an algebra over a commutative ring $\phi$ and let

$$\begin{aligned} p\left( x_1,\,x_2,\,x_3,\,x_4\right) =\sum _{\pi \in S_4}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}, \end{aligned}$$

(6)

be a fixed multilinear polynomial in noncommuting indeterminates $x_i$ over $\phi .$ Here $S_4$ stands for the symmetric group of order 4. Let $\mathcal L$ be a subset of R closed under p,  i.e., $p(\bar{x}_4) \in \mathcal L$ for all $x_1,\, x_2,\, x_3,\, x_4 \in \mathcal L,$ where $\bar{x}_4=(x_1,\,x_2,\,x_3,\,x_4).$ We shall consider a mapping $T{\text {:}}\,\mathcal L \rightarrow R$ satisfying

$$\begin{aligned} T\left( p\left( \bar{x}_4\right) \right)= & {} \sum _{\pi \in S_4}T\left( x_{\pi (1)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} - \sum _{\pi \in S_4}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\nonumber \\&+ \sum _{\pi \in S_4}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( x_{\pi (4)}\right) , \end{aligned}$$

(7)

for all $x_1,\,x_2,\,x_3,\,x_4 \in \mathcal L.$ Let us mention that the idea of considering the expression $[p(\bar{x}_4),\, p(\bar{y}_4)]$ in its proof is taken from the paper of Beidar and Fong [21].

FormalPara Theorem 4

Let $\mathcal L$ be a 8-free Lie subring of R closed under p. If $T{\text {:}}\,\mathcal L \rightarrow R$ is an additive mapping satisfying (7), then there exists $q \in R$ such that $4T(x)=qx+xq$ for all $x \in \mathcal L.$

FormalPara Proof

For any $a \in R$ and $\bar{x}_4 \in \mathcal L^4$, we have

$$\begin{aligned} \left[ p\left( \bar{x}_4\right) ,\,a\right]= & {} p\left( \left[ x_1,\,a\right] ,\,x_2,\,x_3,\,x_4\right) +p\left( x_1,\,\left[ x_2,\,a\right] ,\,x_3,\,x_4\right) \nonumber \\&+p\left( x_1,\,x_2,\,\left[ x_3,\,a\right] ,\,x_4\right) +p\left( x_1,\,x_2,\,x_3,\,\left[ x_4,\,a\right] \right) , \end{aligned}$$

(8)

and therefore

$$\begin{aligned} T\left( \left[ p(\bar{x}_4\right) ,\,a\right] )= & {} \sum _{\pi \in S_4} T\left( \left[ x_{\pi (1)},\,a\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&+ \sum _{\pi \in S_4} T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,a\right] \\&- \sum _{\pi \in S_4} \left[ x_{\pi (1)},\,a \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\\&- \sum _{\pi \in S_4} x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,a\right] \right) x_{\pi (4)} \\&- \sum _{\pi \in S_4} x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,a\right] \\&+ \sum _{\pi \in S_4} \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,a\right] T\left( x_{\pi (4)}\right) \\&+ \sum _{\pi \in S_4} x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,a\right] \right) . \end{aligned}$$

In particular

$$\begin{aligned} T\left( \left[ p\left( \bar{x}_4\right) ,\,p\left( \bar{y}_4\right) \right] \right)= & {} \sum _{\pi \in S_4} T\left( \left[ x_{\pi (1)},\,p\left( \bar{y}_4\right) \right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&+ \sum _{\pi \in S_4} T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,p\left( \bar{y}_4\right) \right] \nonumber \\&- \sum _{\pi \in S_4} \left[ x_{\pi (1)},\,p\left( \bar{y}_4\right) \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)} \nonumber \\&- \sum _{\pi \in S_4} x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,p\left( \bar{y}_4\right) \right] \right) x_{\pi (4)} \nonumber \\&- \sum _{\pi \in S_4} x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,p\left( \bar{y}_4\right) \right] \nonumber \\&+ \sum _{\pi \in S_4} \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,p\left( \bar{y}_4\right) \right] T\left( x_{\pi (4)}\right) \nonumber \\&+ \sum _{\pi \in S_4} x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,p\left( \bar{y}_4\right) \right] \right) . \end{aligned}$$

(9)

Using

$$\begin{aligned} T\left( \left[ x_{\pi (1)},\,p\left( \bar{y}_4\right) \right] \right)= & {} {-}T\left( \left[ p\left( \bar{y}_4\right) ,\,x_{\pi (1)}\right] \right) \nonumber \\= & {} {+}\sum _{\sigma \in S_4} T\left( \left[ x_{\pi (1)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&+ \sum _{\sigma \in S_4} T\left( y_{\sigma (1)}\right) \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \sum _{\sigma \in S_4} \left[ x_{\pi (1)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)} \nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}\nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (1)},\,y_{\sigma (4)}\right] \nonumber \\&+ \sum _{\sigma \in S_4} \left[ x_{\pi (1)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) \nonumber \\&+ \sum _{\sigma \in S_4} y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (4)}\right] \right) ,\nonumber \\ T\left( \left[ x_{\pi (4)},\,p\left( \bar{y}_4\right) \right] \right)= & {} {-}T\left( \left[ p\left( \bar{y}_4\right) ,\,x_{\pi (4)}\right] \right) \nonumber \\= & {} {+}\sum _{\sigma \in S_4} T\left( \left[ x_{\pi (4)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&+ \sum _{\sigma \in S_4} T\left( y_{\sigma (1)}\right) \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \sum _{\sigma \in S_4} \left[ x_{\pi (4)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)} \nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)} \nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \nonumber \\&+ \sum _{\sigma \in S_4} \left[ x_{\pi (4)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) \\ \nonumber&+ \sum _{\sigma \in S_4} y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \right) \end{aligned}$$

and

$$\begin{aligned} T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,p\left( \bar{y}_4\right) \right] \right)= & {} {-}T\left( \left[ p\left( \bar{y}_4\right) ,\, x_{\pi (2)}x_{\pi (3)}\right] \right) \nonumber \\= & {} {+}\sum _{\sigma \in S_4} T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&+ \sum _{\sigma \in S_4} T\left( y_{\sigma (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \sum _{\sigma \in S_4} \left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)} \right] \right) T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)} \nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}\nonumber \\&- \sum _{\sigma \in S_4} y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \nonumber \\&+ \sum _{\sigma \in S_4} \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) \nonumber \\&+ \sum _{\sigma \in S_4} y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \right) \end{aligned}$$

in (9) we have

$$\begin{aligned} T\left( \left[ p\left( \bar{x}_4\right) ,\,p\left( \bar{y}_4\right) \right] \right)= & {} \sum _{\pi \in S_4} \sum _{\sigma \in S_4} \left( T\left( \left[ x_{\pi (1)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\right. \nonumber \\&+\, T\left( y_{\sigma (1)}\right) \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&- \, \left[ x_{\pi (1)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&- \, y_{\sigma (1)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&- \, y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (1)},\,y_{\sigma (4)}\right] x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+ \, \left[ x_{\pi (1)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+ \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (4)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \nonumber \\&+ \, T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \, \left[ x_{\pi (1)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)} \nonumber \\&-\, x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)}\nonumber \\&- \, x_{\pi (1)}T\left( y_{\sigma (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] x_{\pi (4)}\nonumber \\&+ \, x_{\pi (1)}\left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (4)} \nonumber \\&+ \, x_{\pi (1)}y_{\sigma (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}x_{\pi (4)} \nonumber \\&+ \, x_{\pi (1)}y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] x_{\pi (4)}\nonumber \\&- \, x_{\pi (1)}\left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) x_{\pi (4)}\nonumber \\&-\, x_{\pi (1)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \right) x_{\pi (4)}\nonumber \\&- \, x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&+ \, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] T\left( x_{\pi (4)}\right) \nonumber \\&+\, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&+\, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( y_{\sigma (1)}\right) \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}\left[ x_{\pi (4)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)} \nonumber \\&- \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)} \nonumber \\&- \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \nonumber \\&+ \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}\left[ x_{\pi (4)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) \nonumber \\&\left. +\, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \right) \right) . \end{aligned}$$

(10)

If we replace the roles of $\pi$ and $\sigma$ we get

$$\begin{aligned} T\left( \left[ p\left( \bar{x}_4\right) ,\,p\left( \bar{y}_4\right) \right] \right)= & {} \sum _{\pi \in S_4} \sum _{\sigma \in S_4} \left( T\left( \left[ x_{\pi (1)},\,y_{\sigma (1)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right. \nonumber \\&+\, T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (1)}\right] y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&-\, \left[ x_{\pi (1)},\,y_{\sigma (1)} \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&- \, x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] \right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&- \, x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (1)}\right] y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&+ \, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] T\left( x_{\pi (4)}\right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\nonumber \\&+ \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} \nonumber \\&+ \, T\left( y_{\sigma (1)}\right) \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] \nonumber \\&- \, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (1)} \right] T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)} \nonumber \\&-\, y_{\sigma (1)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (4)}\nonumber \\&- \, y_{\sigma (1)}T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] y_{\sigma (4)}\nonumber \\&+ \, y_{\sigma (1)}\left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)} \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}y_{\sigma (4)} \nonumber \\&+ \, y_{\sigma (1)}x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) x_{\pi (4)}y_{\sigma (4)} \nonumber \\&+ \, y_{\sigma (1)}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] y_{\sigma (4)}\nonumber \\&- \, y_{\sigma (1)}\left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] T\left( x_{\pi (4)}\right) y_{\sigma (4)}\nonumber \\&- \, y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)} \nonumber \\&- \, y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (4)}\right] \nonumber \\&+\, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) \nonumber \\&+\, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (4)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+ \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( x_{\pi (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)}x_{\pi (4)},\,y_{\sigma (4)}\right] \nonumber \\&- \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\left[ x_{\pi (1)},\,y_{\sigma (4)} \right] T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)} \nonumber \\&- \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \right) x_{\pi (4)} \nonumber \\&- \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \nonumber \\&+ \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] T\left( x_{\pi (4)}\right) \nonumber \\&\left. +\, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \right) \right) . \end{aligned}$$

(11)

It follows from (10) and (11) that

$$\begin{aligned} 0= & {} \sum _{\pi \in S_4} \sum _{\sigma \in S_4} \left( T\left( \left[ x_{\pi (1)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)} \right. \\&+\, T\left( y_{\sigma (1)}\right) x_{\pi (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}\\&\left. -\, T\left( x_{\pi (1)}\right) y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\\&+\, \sum _{\pi \in S_4} \sum _{\sigma \in S_4} x_{\pi (1)} \left( x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \right) \right. \\&+\, x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}T\left( x_{\pi (4)}\right) \\&- \, x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (4)}T\left( y_{\sigma (4)}\right) \\&-\,2 y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&+\, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( y_{\sigma (4)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&- \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)} T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\\&-\, T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] \right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)}\\&- \, T\left( y_{\sigma (1)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}\right] x_{\pi (4)}\\&-\, y_{\sigma (1)}x_{\pi (2)}x_{\pi (3)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (4)} \\&+\,2 x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (4)}\\&+ \, y_{\sigma (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}x_{\pi (4)} \\&+ \, y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) x_{\pi (2)}x_{\pi (3)}y_{\sigma (4)}x_{\pi (4)}\\&- \, \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}\right] T\left( y_{\sigma (4)}\right) x_{\pi (4)}\\&-\, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \right) x_{\pi (4)}\\&+\, T\left( x_{\pi (2)}x_{\pi (3)}\right) y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)}\\&\left. -\, x_{\pi (2)}x_{\pi (3)}T\left( y_{\sigma (1)}\right) y_{\sigma (2)}y_{\sigma (3)}y_{\sigma (4)}x_{\pi (4)}\right) \\&+\, \sum _{\pi \in S_4} \sum _{\sigma \in S_4} y_{\sigma (1)} \left( y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}T\left( y_{\sigma (4)}\right) \right. \\&- \, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (4)}T\left( x_{\pi (4)}\right) \\&- \, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (4)}\right] \right) \\&- \, x_{\pi (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&- \, T\left( \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \\&- \, T\left( y_{\sigma (2)}y_{\sigma (3)}\right) x_{\pi (1)}y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&- \, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T\left( y_{\sigma (4)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&+ \, y_{\sigma (2)}y_{\sigma (3)}T\left( x_{\pi (1)}\right) y_{\sigma (4)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&+ \, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}y_{\sigma (4)}T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)} \\&\left. +\, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (4)}\right] \right) x_{\pi (4)}\right. \\&\left. -\, y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) y_{\sigma (4)}x_{\pi (4)} \right) \end{aligned}$$

$$\begin{aligned}&+ \, \sum _{\pi \in S_4} \sum _{\sigma \in S_4} \left( T\left( x_{\pi (1)}\right) y_{\sigma (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\right. \nonumber \\&- \, T\left( y_{\sigma (1)}\right) x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&- \, T\left( \left[ x_{\pi (1)},\,y_{\sigma (1)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&+ \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( y_{\sigma (1)}\right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&+ \, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] x_{\pi (4)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) \nonumber \\&-\, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) \nonumber \\&- \, x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) x_{\pi (4)}\nonumber \\&+ \, x_{\pi (1)}y_{\sigma (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&+ \, x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] \right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)} \nonumber \\&-\, x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) y_{\sigma (1)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&-\, \left[ x_{\pi (1)}x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (1)}\right] T\left( x_{\pi (4)}\right) y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&-\,2 y_{\sigma (1)}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&+\, y_{\sigma (1)}T\left( \left[ x_{\pi (1)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+\, y_{\sigma (1)}T\left( x_{\pi (1)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}y_{\sigma (2)}y_{\sigma (3)}\nonumber \\&-\, y_{\sigma (1)}T\left( x_{\pi (1)}\right) y_{\sigma (2)}y_{\sigma (3)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+\,2 y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)} T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\nonumber \\&- \, y_{\sigma (1)}x_{\pi (1)}y_{\sigma (2)}y_{\sigma (3)} T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)} \nonumber \\&- \, y_{\sigma (1)}x_{\pi (1)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) x_{\pi (4)}\nonumber \\&+ \, y_{\sigma (1)}x_{\pi (1)}T\left( x_{\pi (2)}x_{\pi (3)}\right) y_{\sigma (2)}y_{\sigma (3)}x_{\pi (4)}\nonumber \\&+ \, y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}y_{\sigma (2)}y_{\sigma (3)}T\left( x_{\pi (4)}\right) \nonumber \\&- \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( x_{\pi (4)}\right) \nonumber \\&+\, y_{\sigma (1)}x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}T\left( \left[ x_{\pi (4)},\,y_{\sigma (2)}y_{\sigma (3)}\right] \right) \nonumber \\&+\, y_{\sigma (1)}T\left( y_{\sigma (2)}y_{\sigma (3)}\right) x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&\left. - \, y_{\sigma (1)}y_{\sigma (2)}y_{\sigma (3)}T\left( x_{\pi (1)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)} \right) y_{\sigma (4)}, \end{aligned}$$

(12)

for all $x_1,\,x_2,\,x_3,\,x_4,\,y_1,\,y_2,\,y_3,\,y_4 \in \mathcal L.$ Let $s{\text {:}}\,\mathbb Z \rightarrow \mathbb Z$ be a mapping defined by $s(i)=i-4.$ For each $\sigma \in S_4$, the mapping $s^{-1}\sigma s{\text {:}}\,\{5,\,6,\,7,\,8\} \rightarrow \{5,\,6,\,7,\,8\}$ will be denoted by $\overline{\sigma }.$ Let $1\le i<j\le 8$ and define $E^i,\,F^i,\,p^{i,j},\,p^{j,i}{\text {:}}\,{\mathcal L}^8\rightarrow R$ by

$$\begin{aligned} E^i\left( \overline{x}_8\right)= & {} E\left( x_1, \ldots , x_{i-1},\,x_{i+1}, \ldots , x_8\right) , \\ F^i\left( \overline{x}_8\right)= & {} F\left( x_1, \ldots , x_{i-1},\,x_{i+1}, \ldots , x_8\right) , \\ p^{i,j}\left( \overline{x}_8\right)= & {} p^{j,i}\left( \overline{x}_8\right) =\left( x_1, \ldots ,x_{i-1},\,x_{i+1}, \ldots , x_{j-1},\,x_{j+1},\ldots , x_8\right) , \end{aligned}$$

where $\overline{x}_8=(x_1, \ldots , x_8)\in {\mathcal L}^8.$ Writing $x_{4+i}$ instead of $y_i,\,i=1,\,2,\,3,\,4$ in the (12) we can express this relation as

$$\begin{aligned} \sum _{i=1}^8E_i^i\left( \overline{x}_8\right) x_i + \sum _{j=1}^8x_jF_j^j\left( \overline{x}_8\right) =0. \end{aligned}$$

For example,

$$\begin{aligned} E_4^4\left( \overline{x}_8\right)= & {} \sum _{\begin{array}{c} \pi \in S_{4}\\ {{\pi }}(4)=4 \end{array}} \sum _{\overline{\sigma } \in S_{4}} \left( T\left( \left[ x_{\pi (1)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}\right. \nonumber \\&+\, T\left( x_{\overline{\sigma }(5)}\right) x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)} \nonumber \\&\left. -\, T\left( x_{\pi (1)}\right) x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}\right) , \end{aligned}$$

(13)

and

$$\begin{aligned} F_1^1\left( \overline{x}_8\right)= & {} \sum _{\begin{array}{c} \pi \in S_{4}\\ {{\pi }}(1)=1 \end{array}}\sum _{\overline{\sigma } \in S_{4}} \left( x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T\left( \left[ x_{\pi (4)},\,x_{\overline{\sigma }(8)}\right] \right) \right. \nonumber \\&+ \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}T\left( x_{\pi (4)}\right) \nonumber \\&- \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\pi (4)}T\left( x_{\overline{\sigma }(8)}\right) \nonumber \\&-\,2 x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&+\, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T\left( x_{\overline{\sigma }(8)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&-\, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\nonumber \\&-\, T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&-\, T\left( x_{\overline{\sigma }(5)}\right) \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\right] x_{\pi (4)}\nonumber \\&-\, x_{\overline{\sigma }(5)}x_{\pi (2)}x_{\pi (3)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (4)} \nonumber \\&+\,2 x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&+ \, x_{\overline{\sigma }(5)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right] \right) x_{\overline{\sigma }(8)}x_{\pi (4)} \nonumber \\&+ \, x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&- \, \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right] T\left( x_{\overline{\sigma }(8)}\right) x_{\pi (4)}\nonumber \\&-\, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(8)}\right] \right) x_{\pi (4)}\nonumber \\&+ \, T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&\left. - \, x_{\pi (2)}x_{\pi (3)}T\left( x_{\overline{\sigma }(5)}\right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)} \right) , \end{aligned}$$

(14)

for all $\overline{x}_8 \in \mathcal L^8.$ Since $\mathcal L$ is 8-free, it follows that the functional identity (13) has only a standard solution. In particular, there exist mappings $p_{4,j}{\text {:}}\,\mathcal L^6 \rightarrow R,\,j=1,\,2,\,3,\,5,\,6,\,7,\,8$ and $\lambda _{4}{\text {:}}\,\mathcal L^7\rightarrow C(\mathcal L)$ such that

$$\begin{aligned} E_4^4\left( \overline{x}_8\right) = \sum _{\begin{array}{c} j=1\\ j\ne 4 \end{array}}^8x_jp_{4,j}^{4,j}\left( \overline{x}_8\right) +\lambda _4^4\left( \overline{x}_8\right) , \end{aligned}$$

for all $\overline{x}_8\in \mathcal L^8.$ Hence

$$\begin{aligned}&\sum _{\begin{array}{c} \pi \in S_{4}\\ {{\pi }}(4)=4 \end{array}}\sum _{\overline{\sigma } \in S_{4}} \left( T\left( \left[ x_{\pi (1)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}\right. \nonumber \\&+\, T\left( x_{\overline{\sigma }(5)}\right) x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)} \nonumber \\&\left. - \, T\left( x_{\pi (1)}\right) x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}\right) x_{\pi (3)} -\sum _{\begin{array}{c} j=1\\ j\ne 4 \end{array}}^8x_jp_{4,j}^{4,j}\left( \overline{x}_8\right) \in C(\mathcal L), \end{aligned}$$

for all $\overline{x}_8\in \mathcal L^8.$ Let us write

$$\begin{aligned} E_{4,3}^{4,3}\left( \overline{x}_8\right)= & {} \sum _{\begin{array}{c} \pi \in S_{4}\\ {{\pi }}(4)=4 \end{array}}\sum _{\overline{\sigma } \in S_{4}} \left( T\left( \left[ x_{\pi (1)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\right. \nonumber \\&+\, T\left( x_{\overline{\sigma }(5)}\right) x_{\pi (1)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} \nonumber \\&\left. -\, T\left( x_{\pi (1)}\right) x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\right) x_{\pi (2)}, \end{aligned}$$

for all $\overline{x}_8\in {\mathcal L}^8.$ Again using that $\mathcal L$ is 8-free, this identity has only a standard solution. Hence there exist mappings $p_{3,j}{\text {:}}\,\mathcal L^5\rightarrow R,\,j=1,\,2,\,5,\,6,\,7,\,8$ such that

$$\begin{aligned} E_{4,3}^{4,3}\left( \overline{x}_8\right) -\sum _{\begin{array}{c} j=1\\ j\ne 3,4 \end{array}}^8x_jp_{4,3,j}^{4,3,j}\left( \overline{x}_8\right) \in C(\mathcal L). \end{aligned}$$

We continue with the same procedure as above. Since $\mathcal L$ is 8-free, after a finite number of steps, we arrive at

$$\begin{aligned} T([x,\,y])+T(y)x-T(x)y=xf(y)+yg(x)+\lambda (x,\,y), \end{aligned}$$

(15)

for all $x,\,y \in \mathcal L,$ where $f,\,g{\text {:}}\, \mathcal L \rightarrow R$ and $\lambda {\text {:}}\, \mathcal L^2 \rightarrow C(\mathcal L).$ On the other hand from (14) and using the theory of functional identities a few more time, we can conclude that

$$\begin{aligned} T([x,\,y])-xT(y)+yT(x)=h(y)x+k(x)y+\mu (x,\,y), \end{aligned}$$

(16)

for all $x,\,y \in \mathcal L,$ where $h,\,k{\text {:}}\, \mathcal L \rightarrow R$ and $\mu {\text {:}}\, \mathcal L^2 \rightarrow C(\mathcal L).$ If we replace the roles of denotations x and y in (15) and compare so obtained identities, we arrive at $0=x(f(y)+g(y))+y(g(x)+f(x))+\lambda (x,\,y)+\lambda (y,\,x).$ Since $\mathcal L$ is a 8-free subset of R, we have $f(x)={-}g(x)$ for all $x \in \mathcal L$ and $\lambda (x,\,y)={-}\lambda (y,\,x).$ Equation (15) can now be rewritten as

$$\begin{aligned} T([x,\,y])+T(y)x-T(x)y=xf(y)-yf(x)+\lambda (x,\,y). \end{aligned}$$

(17)

Similar if we replace the roles of denotations x and y in (16) and compare so obtained identities, we arrive at $0=(h(y)+k(y))x+(h(x)+g(x))y+\mu (x,\,y)+\mu (y,\,x).$ Since $\mathcal L$ is a 8-free subset of R, we have $h(x)={-}k(x)$ for all $x \in \mathcal L$ and $\mu (x,\,y)={-}\mu (y,\,x).$ Equation (16) can now be rewritten as

$$\begin{aligned} T([x,\,y])-xT(y)+yT(x)=h(y)x-h(x)y+\mu (x,\,y). \end{aligned}$$

(18)

Using (17) and (18), there exist $r \in R$ and $\mu ,\,\mu ^{\prime }{\text {:}}\,\mathcal L \rightarrow C(\mathcal L)$ such that

$$\begin{aligned} T(x)-h(x)= & {} xr+\mu (x), \nonumber \\ T(x)-f(x)= & {} rx+\mu ^{\prime }(x), \end{aligned}$$

(19)

for all $x \in \mathcal L.$ Now, if we use (17) and (18) in (12), it is easy to see, that this functional identity is of the form $\sum _{j=1}^8x_jH_j^j(\overline{x}_8)=0$ where in particular

$$\begin{aligned} H_1^1\left( \overline{x}_8\right)= & {} \sum _{\pi \in S_4}\sum _{\overline{\sigma } \in S_4} \left( T\left( x_{\overline{\sigma }(5)}\right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\right. \nonumber \\&+\, f\left( x_{\overline{\sigma }(5)}\right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&-\, T\left( x_{\overline{\sigma }(5)}\right) x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&- \, f\left( x_{\overline{\sigma }(5)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)} x_{\overline{\sigma }(8)}\\&+\, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}h\left( x_{\overline{\sigma }(8)}\right) x_{\pi (4)} \nonumber \\&-\, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}h\left( x_{\pi (4)}\right) x_{\overline{\sigma }(8)}\\&+ \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)} \mu \left( x_{\pi (4)},\,x_{\overline{\sigma }(8)}\right) \nonumber \\&-\,2 x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\\&+ \, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T\left( x_{\overline{\sigma }(8)}\right) x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\nonumber \\&-\, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)} T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\\&-\, T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&+\,2 x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (4)}\\&+\, x_{\overline{\sigma }(5)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right] \right) x_{\overline{\sigma }(8)}x_{\pi (4)} \nonumber \\&+ \, x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(8)}x_{\pi (4)}\\&- \, \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right] T\left( x_{\overline{\sigma }(8)}\right) x_{\pi (4)}\nonumber \\&- \, x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(8)}\right] \right) x_{\pi (4)}\\&+ \, T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\overline{\sigma }(5)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\nonumber \\&- \, x_{\pi (2)}x_{\pi (3)}T\left( x_{\overline{\sigma }(5)}\right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}x_{\pi (4)}\\&- \, x_{\overline{\sigma }(5)}x_{\pi (2)}x_{\pi (3)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)}x_{\pi (4)} \nonumber \\&+ \, x_{\pi (2)}x_{\pi (3)}T\left( x_{\overline{\sigma }(5)}\right) x_{\pi (4)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\\&+ \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}x_{\pi (4)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\overline{\sigma }(8)} \nonumber \\&- \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}T\left( \left[ x_{\pi (4)},\,x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right] \right) x_{\overline{\sigma }(8)}\\&- \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}T\left( x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}\right) x_{\pi (4)}x_{\overline{\sigma }(8)}\nonumber \\&+\, x_{\overline{\sigma }(5)}T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)} x_{\overline{\sigma }(8)}\\&+ \, T\left( \left[ x_{\pi (2)}x_{\pi (3)},\,x_{\overline{\sigma }(5)}\right] \right) x_{\pi (4)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\nonumber \\&\left. -\, T\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\overline{\sigma }(5)}x_{\pi (4)}x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\right. \\&\left. - \, x_{\pi (2)}x_{\pi (3)}x_{\overline{\sigma }(5)}T\left( x_{\pi (4)}\right) x_{\overline{\sigma }(6)}x_{\overline{\sigma }(7)}x_{\overline{\sigma }(8)}\right) , \end{aligned}$$

for all $x_2,\,x_3,\, x_4,\,x_5,\,x_6,\,x_7 ,\, x_8 \in \mathcal L.$ Consequently after using the theory of functional identities a few more times, there exist $r^{\prime } \in R$ and a mapping $\mu ^{\prime \prime }{\text {:}}\, \mathcal L \rightarrow C(\mathcal L)$ such that

$$\begin{aligned} T(x)+f(x)=xr^{\prime }+\mu ^{\prime \prime }(x), \end{aligned}$$

for all $x \in \mathcal L.$ On the other hand by (19), we have

$$\begin{aligned} T(x)-f(x)=rx+\mu ^{\prime }(x), \end{aligned}$$

for all $x \in \mathcal L.$ Thus

$$\begin{aligned} 2T(x)=xr^{\prime }+rx+\mu ^{\prime }(x)+\mu ^{\prime \prime }(x). \end{aligned}$$

By (7) we arrive at

$$\begin{aligned} 0= & {} \sum _{\pi \in S_4} x_{\pi (1)}\left( r^{\prime }x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}-rx_{\pi (2)}x_{\pi (3)}x_{\pi (4)}-x_{\pi (2)}x_{\pi (3)}r^{\prime }x_{\pi (4)}\right. \\&\left. +\,x_{\pi (2)}x_{\pi (3)}rx_{\pi (4)}-(\mu ^{\prime }+\mu ^{\prime \prime })\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}\right) \\&+\,x_{\pi (2)}\left( (\mu ^{\prime }+\mu ^{\prime \prime })\left( x_{\pi (1)}\right) x_{\pi (3)}x_{\pi (4)}\right) \\&+\,\mu ^{\prime }\left( x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\right) +\mu ^{\prime \prime }\left( x_{\pi (1)}x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}\right) . \end{aligned}$$

Note that this implies $\mu ^{\prime }(x)+\mu ^{\prime \prime }(x)=0$ and

$$\begin{aligned} 0= & {} \sum _{\pi \in S_4,\,\pi (1)=1 } r^{\prime }x_{\pi (2)}x_{\pi (3)}x_{\pi (4)}-rx_{\pi (2)}x_{\pi (3)}x_{\pi (4)}-x_{\pi (2)}x_{\pi (3)}r^{\prime }x_{\pi (4)} \nonumber \\&+\,x_{\pi (2)}x_{\pi (3)}rx_{\pi (4)}-(\mu ^{\prime }+\mu ^{\prime \prime })\left( x_{\pi (2)}x_{\pi (3)}\right) x_{\pi (4)}. \end{aligned}$$

Using theory one more time, we get

$$\begin{aligned} 0= & {} \sum _{\pi \in S_4,\,\pi (1)=1,\,\pi (4)=4} r^{\prime }x_{\pi (2)}x_{\pi (3)}-rx_{\pi (2)}x_{\pi (3)}-x_{\pi (2)}x_{\pi (3)}r^{\prime } \\&+x_{\pi (2)}x_{\pi (3)}r-(\mu ^{\prime }+\mu ^{\prime \prime })\left( x_{\pi (2)}x_{\pi (3)}\right) . \end{aligned}$$

Hence there exist $q,\,s \in R$ and $\lambda _q,\,\lambda _s{\text {:}}\,\mathcal L \rightarrow C(\mathcal L)$ such that

$$\begin{aligned} r^{\prime }x-rx= & {} xs+\lambda _s(x), \\ -xr^{\prime }+xr= & {} qx+\lambda _q(x). \end{aligned}$$

Now we can conclude that $xs=sx,\,\lambda _s(x)=0$ and $\lambda _s(xy)=0.$ Last equation can now be rewritten as $(r^{\prime }-r-s)x=0.$ Now it follows $r^{\prime }-r=s$ and so $r^{\prime }-r\in C(\mathcal L).$ Thus $[r,\,x]=[r^{\prime },\,x]$ for all $x \in \mathcal L,$ which implies

$$\begin{aligned} 2T(x)= & {} rx+xr^{\prime }, \\ 2T(x)= & {} r^{\prime }x+xr. \end{aligned}$$

Consequently, $4T(x)=(r+r^{\prime })x+x(r+r^{\prime })$ for all $x \in \mathcal L.$ Thereby the proof is completed. $\square$

We are now in the position to prove Theorem  3.

FormalPara Proof

The complete linearization of (5) gives us (7). First suppose that R is not a PI ring. According to Theorem 4, there exist $q\in Q_\mathrm{mr}$ such that $4T(x)=qx+xq$ for all $x \in R.$ Since $qx+xq \subset R$ for all $x \in R$, it follows from the end of the proof of ([14], Theorem 2.1) that $q \in Q_\mathrm{s}.$

Now suppose that R is a PI ring. It is well-known that in this case R has a nonzero center (see [22]). Let c be a nonzero central element. Pick any $x \in R$ and set $x_1=x_2=x_3=cx$ and $x_4=x$ in (7). Hence we obtain

$$\begin{aligned} 4T\left( c^3x^4\right)= & {} T(x)c^3x^3+3T(cx)c^2x^3-2cxT\left( c^2x^2\right) x \nonumber \\&-2c^2xT\left( cx^2\right) x+3c^2x^3T(cx)+c^3x^3T(x). \end{aligned}$$

Next, setting $x_1=x_2=x_3=c$ and $x_4=x^4$ in (7), we arrive at

$$\begin{aligned} 4T\left( c^3x^4\right)= & {} 2c^3T\left( x^4\right) +3c^2T(c)x^4-cx^4T\left( c^2\right) \\ \nonumber&-2c^2T\left( cx^4\right) -cT\left( c^2\right) x^4+3c^2x^4T(c), \end{aligned}$$

for all $x \in R.$ Comparing last two identities and using equation $T(x^4)=T(x)x^3-xT(x^2)x+x^3T(x)$, we get

$$\begin{aligned} 0= & {} -c^3T(x)x^3-c^3x^3T(x)+2c^3xT\left( x^2\right) x \nonumber \\&+\,3T(cx)c^2x^3+3c^2x^3T(cx)-2c^2xT\left( cx^2\right) x \nonumber \\&-\,2cxT\left( c^2x^2\right) x-3c^2T(c)x^4-3c^2x^4T(c) \nonumber \\&+\,cx^4T\left( c^2\right) +cT\left( c^2\right) x^4+2c^2T\left( cx^4\right) , \end{aligned}$$

(20)

for all $x \in R.$ Setting $x_1=x_2=c$ and $x_3=x_4=x$ in (7), we get

$$\begin{aligned} 12T\left( c^2x^2\right)= & {} 6T(x)c^2x+6cT(c)x^2-2c^2T\left( x^2\right) \nonumber \\&-4cxT(cx)-4cT(cx)x-2xT\left( c^2\right) x+6cx^2T(c)+6c^2xT(x),\nonumber \\ \end{aligned}$$

(21)

for all $x \in R.$ Now setting $x_1=x_2=x_3=x$ and $x_4=cx$ in (7), we get

$$\begin{aligned} 4T\left( cx^4\right)= & {} T(cx)x^3+3T(x)cx^3-2cxT\left( x^2\right) x \nonumber \\&-2xT\left( cx^2\right) x+x^3T(cx)+3cx^3T(x), \end{aligned}$$

(22)

for all $x \in R.$ Now putting (21) and (22) into (20), we get

$$\begin{aligned} 0= & {} 3c^2T(x)x^3+3c^2x^3T(x)+21cT(cx)x^3+21cx^3T(cx) \nonumber \\&-18cxT\left( cx^2\right) x+8c^2xT\left( x^2\right) x-18cT(c)x^4-18cx^4T(c) \nonumber \\&+6x^4T\left( c^2\right) +6T\left( c^2\right) x^4+2x^2T\left( c^2\right) x^2+4cx^2T(cx)x+4cxT(cx)x^2 \nonumber \\&-6c^2xT(x)x^2-6c^2x^2T(x)x-6cxT(c)x^3-6cx^3T(c)x. \end{aligned}$$

(23)

Substituting x by c in the last equation, we obtain

$$\begin{aligned} T\left( c^3\right) =4cT\left( c^2\right) -3c^2T(c). \end{aligned}$$

After complete linearization of (23), setting $x_1=x,\,x_2=x_3=x_4=c$ and using $T(c^3)=4cT(c^2)-3c^2T(c)$, we get

$$\begin{aligned} 12T\left( c^2x\right)= & {} 22cT(cx)-2c^2T(x)+13T\left( c^2\right) x+13xT\left( c^2\right) \nonumber \\&-17cxT(c)-17cT(c)x. \end{aligned}$$

(24)

Substituting cx for x in the last relation, we obtain

$$\begin{aligned} 12T\left( c^3x\right)= & {} 22cT\left( c^2x\right) -2c^2T(cx)+13cT\left( c^2\right) x+13cxT\left( c^2\right) \nonumber \\&-17c^2xT(c)-17c^2T(c)x. \end{aligned}$$

(25)

Putting (24) into the last equation, we get

$$\begin{aligned} 72T\left( c^3x\right)= & {} 230c^2T(cx)+221cT\left( c^2\right) x+221cxT\left( c^2\right) \nonumber \\&-289c^2xT(c)-289c^2T(c)x-22c^3T(x). \end{aligned}$$

(26)

Next, setting $x_1=x_2=x_3=c$ and $x_4=x$ in (7), we have

$$\begin{aligned} 4T\left( c^3x\right)= & {} 2c^3T(x)+3c^2T(c)x+3c^2xT(c)-cxT\left( c^2\right) \nonumber \\&-\,cT\left( c^2\right) x-2c^2T(cx). \end{aligned}$$

(27)

Comparing Eqs. (26) and (27), we see that

$$\begin{aligned} 266cT(cx)= & {} 58c^2T(x)-239T\left( c^2\right) x-239xT\left( c^2\right) \nonumber \\&+\,343cT(c)x+343cxT(c). \end{aligned}$$

(28)

Setting c instead of x in the last equation, we obtain

$$\begin{aligned} T\left( c^2\right) =cT(c). \end{aligned}$$

(29)

Putting (29) into (28), we obtain

$$\begin{aligned} 133T(cx)=29cT(x)+52T(c)x+52xT(c). \end{aligned}$$

(30)

Substituting $c^2$ for c in the last equation and using $T(c^2)=cT(c)$, we obtain

$$\begin{aligned} 133T\left( c^2x\right) =29c^2T(x)+52cT(c)x+52cxT(c). \end{aligned}$$

(31)

Setting cx instead of x in the (30), we obtain

$$\begin{aligned} 133T\left( c^2x\right) =29cT(cx)+52cT(c)x+52cxT(c). \end{aligned}$$

(32)

Comparing the last two identities leads to

$$\begin{aligned} T(cx)=cT(x), \end{aligned}$$

(33)

for all $x \in R.$ Now using last equation into (30), we obtain

$$\begin{aligned} 4cT(x)=2T(c)x+2xT(c). \end{aligned}$$

(34)

for all $x \in R.$ Now let F be the field of fractions of R. Enlarge R to the ring RF, noting that any element of RF can be written in the form $rc^{-1}.$ Then D can be extended to RF by defining

$$\begin{aligned} T\left( c^{-1}x\right) =c^{-1}T(x), \end{aligned}$$

for all $x \in R.$ This is well-defined. Namely, if $xc^{-1}=yd^{-1},$ then $cy=dx,$ whence by (33)

$$\begin{aligned} cT(y)=T(cy)=T(dx)=dT(x), \end{aligned}$$

which implies $c^{-1}T(x)=d^{-1}T(y).$ Since RF has the identity element $1=cc^{-1},$ we obtain from the Eq. (34) that

$$\begin{aligned} 4T(x)=2c^{-1}T(c)x+2xc^{-1}T(c), \end{aligned}$$

(35)

for all $x \in R.$ Now $2c^{-1}T(c)$ can be written as q,  so we get

$$\begin{aligned} 4T(x)=qx+xq, \end{aligned}$$

(36)

for all $x \in R$ and some fixed element $q\in Q_\mathrm{s}.$ Thereby the proof is completed. $\square$

In our recent paper [1], one can find the following result.

FormalPara Theorem 5

Let R be a prime ring with $\mathrm{char}(R)=0$ or $\mathrm{char}(R)>4$ and let $D{\text {:}}\,R\rightarrow R$ be an additive mapping satisfying the relation

$$\begin{aligned} D\left( x^{4}\right) =D(x)x^{3}+xD\left( x^{2}\right) x+x^{3}D(x), \end{aligned}$$

(37)

for all $x\in R.$ In this case, D is a derivation.

The next result contains Theorem 3 as well as Theorem 5.

FormalPara Theorem 6

Let R be a prime ring with $\mathrm{char}(R)=0$ or $\mathrm{char}(R)>4$ and let $S,\,T{\text {:}}\,R\rightarrow R$ be additive mappings satisfying the relations

$$\begin{aligned} S\left( x^{4}\right)= & {} S(x)x^{3}+xT\left( x^{2}\right) x+x^{3}S(x), \end{aligned}$$

(38)

$$\begin{aligned} T\left( x^{4}\right)= & {} T(x)x^{3}+xS\left( x^{2}\right) x+x^{3}T(x), \end{aligned}$$

(39)

for all $x\in R.$ In this case, S and T are of the form $8S(x)=4D(x)+qx+xq,\,8T(x)=4D(x)-qx-xq$ for all $x\in R$ and some fixed $q\in Q_\mathrm{s},$ where $D{\text {:}}\,R\rightarrow R$ is a derivation.

FormalPara Proof

Combining (38) and (39), we obtain

$$\begin{aligned} D\left( x^{4}\right) =D(x)x^{3}+xD\left( x^{2}\right) x+x^{3}D(x), \end{aligned}$$

(40)

for all $x\in R,$ where D stands for $S+T.$ According to Theorem  5, D is a derivation. Subtracting the relation (39) from the relation (38) one obtains

$$\begin{aligned} G\left( x^{4}\right) =G(x)x^{3}-xG\left( x^{2}\right) x+x^{3}G(x), \end{aligned}$$

(41)

for all $x\in R,$ where G denotes $S-T.$ Since all the assumptions of Theorem 3 are fulfilled, one can conclude that G is of the form $4G(x)=qx+xq$ for all $x\in R$ and some fixed $q\in Q_\mathrm{s}.$ Since $2S(x)=D(x)+G(x),\,2T(x)=D(x)-G(x)$ it follows $8S(x)=4D(x)+4G(x)=4D(x)+qx+xq,\,8T(x)=4D(x)-4G(x)=4D(x)-qx-xq,$ which completeness the proof of the theorem. $\square$

Stachó and Zalar [23, 24] investigated bicircular projections on the $C^{*}$-algebra $\mathcal {L}(H),$ where H is a complex Hilbert space. According to Proposition 3.4 in [23], every bicircular projection $P{\text {:}}\,\mathcal {L}(H)\rightarrow \mathcal {L}(H),$ where H is a complex Hilbert space, satisfies the functional equation

$$\begin{aligned} P(xyx)=P(x)yx-xP(y^{*})^{*}x+xyP(x), \end{aligned}$$

(42)

for all pairs $x,\,y\in \mathcal {L}(H).$ Bicircular projections and related functional equations have been extensively investigated during the last few years (see [16–18, 23–36]). Fošner and Ilišević [27] investigated the above functional equation on 2-torsion-free semiprime $^{*}$-ring. They expressed the solution of the functional equation (42) in terms of derivation and so-called double centralizers (see also [34]). Fošner and Vukman [17] considered the following system of functional equations

$$\begin{aligned} P\left( x^{3}\right)= & {} P(x)x^{2}+xQ(x^{*})^{*}x+x^{2}P(x), \\ Q\left( x^{3}\right)= & {} Q(x)x^{2}+xP(x^{*})^{*}x+x^{2}Q(x), \end{aligned}$$

where P and Q are additive mappings, which map a prime $^{*}$-ring with characteristic different from two into itself (see [18] for a generalization). We conclude with our last result.

FormalPara Theorem 7

Let R be a prime $^{*}$-ring with $\mathrm{char}(R)=0$ or $\mathrm{char}(R)>4$ and let $P,\,Q{\text {:}}\,R\rightarrow R$ be additive mappings satisfying the relations

$$\begin{aligned} P\left( x^{4}\right)= & {} P(x)x^{3}+xQ\left( x^{*2}\right) ^{*}x+x^{3}P(x), \\ Q\left( x^{4}\right)= & {} Q(x)x^{3}+xP\left( x^{*2}\right) ^{*}x+x^{3}Q(x), \end{aligned}$$

for all $x\in R.$ In this case, P and Q are of the form

$$\begin{aligned} 16P(x)= & {} 4(d(x)+g(x))+(p+q)x+x(p+q), \\ 16Q(x)= & {} 4(d(x)-g(x))+(q-p)x+x(q-p), \end{aligned}$$

for all $x{\in } R,$ where $d,\,g{\text {:}}\,R{\rightarrow } R$ are derivations and $p,\,q\in Q_\mathrm{s}$ are some fixed elements. Besides $d(x^{*})^{*}={-}d(x),\, g(x^{*})^{*}=g(x)$ for all $x{\in } R,$ and $p^{*}=p,\,q^{*}={-}q.$

FormalPara Proof

The proof goes through in three steps.

First step. Let us assume that $P=Q$ and let F denote P. In this case, we have the relation

$$\begin{aligned} F\left( x^{4}\right) =F(x)x^{3}+xF\left( x^{*2}\right) ^{*}x+x^{3}F(x), \end{aligned}$$

(43)

for all $x\in R.$ It is our aim to prove that F is of the form

$$\begin{aligned} 8F(x)=4d(x)+qx+xq, \end{aligned}$$

(44)

for all $x\in R,$ where d is a derivation of R and q is a fixed element from $Q_\mathrm{s}.$ Besides, $d(x^{*})^{*}=d(x)$ for all $x\in R,$ and $q^{*}={-}q.$ Let us introduce mappings $d{\text {:}}\,R\rightarrow R$ and $f{\text {:}}\,R\rightarrow R$ by

$$\begin{aligned} d(x)= & {} F(x)+F(x^{*})^{*},\nonumber \\ f(x)= & {} F(x)-F(x^{*})^{*}, \end{aligned}$$

(45)

for all $x \in R.$ Now we have

$$\begin{aligned} d(x^{*})^{*}=(F(x^{*})+F(x)^{*})^{*}=F(x^{*})^{*}+F(x)=d(x), \end{aligned}$$

for all $x\in R.$ Applying the relation (43), one obtains easily that

$$\begin{aligned} d\left( x^{4}\right) =d(x)x^{3}+xd\left( x^{2}\right) x+x^{3}d(x), \end{aligned}$$

(46)

and

$$\begin{aligned} f\left( x^{4}\right) =f(x)x^{3}-xf\left( x^{2}\right) x+x^{3}f(x), \end{aligned}$$

(47)

is fulfilled for all $x\in R.$ Now it follows from the relation (46) and Theorem 5 that d is a derivation. On the other hand, one can conclude from the relation (47) and Theorem 3 that f is of the form $4f(x)=qx+xq$ for all $x\in R$ and some fixed $q\in Q_\mathrm{s}.$ We have therefore

$$\begin{aligned} 4F(x)-4F(x^{*})^{*}=qx+xq, \end{aligned}$$

(48)

for all $x\in R.$ Putting in the above relation $x^{*}$ for x we obtain $4F(x^{*})-4F(x)^{*}=qx^{*}+xq^{*},\,x\in R,$ which gives

$$\begin{aligned} 4F(x^{*})^{*}-4F(x)=q^{*}x+xq^{*}, \end{aligned}$$

for all $x\in R.$ Combining the above relation with the relation (48), we obtain $(q+q^{*})x+x(q+q^{*})=0$ or all $x\in R,$ hence it follows after some calculation because of primeness of R that $q^{*}={-}q.$ Combining the relation (45) with the relation (48), we obtain $8F(x)=4d(x)+qx+xq,$ which completes the proof of the first step.

Second step. Let us assume that $Q=P$ and let H denote P. In this case, we have the relation

$$\begin{aligned} H\left( x^{4}\right) =H(x)x^{3}-xH\left( x^{*2}\right) ^{*}x+x^{3}H(x), \end{aligned}$$

for all $x\in R.$ In this case, H is of the form

$$\begin{aligned} 8H(x)=4g(x)+px+xp, \end{aligned}$$

(49)

for all $x\in R,$ where g is a derivation of R,  and p is a fixed element from $Q_\mathrm{s}.$ Besides, $g(x^{*})^{*}={-}g(x)$ for all $x\in R,$ and $p^{*}=p.$ The proof of the second step will be omitted since it goes through using the same arguments as in the proof of the first step.

Third step. We are ready for the proof of general case. We have therefore the relations

$$\begin{aligned} P\left( x^{4}\right) =P(x)x^{3}+xQ\left( x^{*2}\right) ^{*}x+x^{3}P(x),\quad x\in R, \end{aligned}$$

(50)

and

$$\begin{aligned} Q\left( x^{4}\right) =Q(x)x^{3}+xP\left( x^{*2}\right) ^{*}x+x^{3}Q(x),\quad x\in R. \end{aligned}$$

(51)

Combining the relation (50) with the relation (51), we obtain

$$\begin{aligned} F\left( x^{4}\right) =F(x)x^{3}+xF\left( x^{*2}\right) ^{*}x+x^{3}F(x),\quad x\in R, \end{aligned}$$

(52)

where F stands for $P+Q.$ On the other hand, subtracting the relation (52) from the relation (51), we arrive at

$$\begin{aligned} H\left( x^{4}\right) =H(x)x^{3}-xH\left( x^{*2}\right) ^{*}x+x^{3}H(x),\quad x\in R, \end{aligned}$$

where H denotes $P-Q.$ Now according to (44) and (49), we have

$$\begin{aligned} 8P(x)+8Q(x)= & {} 4d(x)+qx+xq, \end{aligned}$$

(53)

$$\begin{aligned} 8P(x)-8Q(x)= & {} 4g(x)+px+xp. \end{aligned}$$

(54)

From (53) and (54), one obtains

$$\begin{aligned} 16P(x)=4(d(x)+g(x))+(p+q)x+x(p+q), \end{aligned}$$

and

$$\begin{aligned} 16Q(x)=4(d(x)-g(x))+(q-p)x+x(q-p), \end{aligned}$$

which completes the proof of the theorem. $\square$