On certain functional equation in prime rings

Introduction

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 ∈ R , n x = 0 implies x = 0 . The commutator x y − y x will be denoted by [ x , y ] . A ring R is prime if for a , b ∈ R , a R b = ( 0 ) implies that either a = 0 or b = 0 and is semiprime in case a R a = ( 0 ) implies a = 0 . We denote by Q m r , Q 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 [1], Chapter 2). An additive mapping D : R → R is called a derivation if D ( x y ) = D ( x ) y + x D ( y ) holds for all pairs x , y ∈ R and is called a Jordan derivation in case D ( x 2 ) = D ( x ) x + x D ( x ) is fulfilled for all x ∈ R . A derivation D : R → R is inner in case D is of the form D ( x ) = [ a , x ] for all x ∈ R and some fixed a ∈ R . Every derivation is Jordan derivation. The converse is in general not true. A classical result of Herstein [2] 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 [3]. In [4], one can find a generalization of Herstein theorem. Cusack [5] generalized Herstein theorem to 2-torsion free semiprime rings (see [6] for an alternative proof). Herstein theorem has been fairly generalized by Beidar et al. [7]. For results related to Herstein theorem, we refer to [8,9, 10,11]. We proceed with the following result proved by Brešar [12] (see [13] for a generalization).

An additive mapping satisfying the relation (1) on an 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, 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).

We proceed with the following functional equation:

which appears naturally in the proof of Theorem 2 in [16]. One can easily prove that in case we have an additive mapping F : R → R , where R is 2-torsion free semiprime ring, satisfying the relation (3) for all pairs x , y ∈ R , then F is of the form 2 F ( x ) = D ( x ) + a x + x a , where D : R → R is a derivation and a ∈ R some fixed element (see [16] for the details). In [16], one can find the following conjecture.

By our knowledge, the aforementioned conjecture is still an open question. The substitution y = x in (1), (2), and (3) gives

and

The relation (4) has been considered in [7] (actually, much more general situation has been considered). A result related to (5) can be found in [15]. It is our aim in this paper to prove the following result, which is related to the aforementioned conjecture.

Main results

As the main tool in this paper, we use the theory of functional identities (Brešar-Beidar-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 [17] for the introductory account on the theory of functional identities, where Brešar presents this theory and its applications to a wider audience and to [18] for the full treatment of this theory.

Let R be an algebra over a commutative ring ϕ and let

be a fixed multilinear polynomial in noncommuting indeterminates x i over ϕ . Here, S 3 stands for the symmetric group of order 3. Let ℒ be a subset of R closed under p , i.e., p ( x ¯ 3 ) ∈ ℒ for all x 1 , x 2 , x 3 ∈ ℒ , where x ¯ 3 = ( x 1 , x 2 , x 3 ) . We shall consider a mapping D : ℒ → R satisfying

for all x 1 , x 2 , x 3 ∈ ℒ . Let us mention that the idea of considering the expression [ p ( x ¯ 3 ) , p ( y ¯ 3 ) ] in its proof is taken from [19]. For the proof of Theorem 4, we need Theorem 5, which might be of independent interest.