FUNCTIONAL IDENTITIES WITH r-INDEPENDENT COEFFICIENTS

Abstract

derivation provided that dð½x; yÞ ¼ ½dðxÞ; y þ ½x; dðyÞ for all x; y 2 A. The study of Lie maps of associative rings goes back to Hua (1951). At his 1961 AMS Hour Talk [32, pp. 528-529] Herstein formulated a program of studying Lie maps in associative (prime or simple) rings. Over the next 30 years Martindale and some of his students solved a number of Herstein's problems in the class of prime rings under additional assumption on existence of certain idempotents (see [36] and references over there).In 1993 Bresˇar [19] solved Herstein's problem on the description of a Lie isomorphism a : A ! B of prime rings A and B without assuming the existence of idempotents via the solution of the following FI: ½Bðx; xÞ; x ¼ 0 for all x 2 A, where B : A Â A ! A is a biadditive map (in the case of Lie isomorphisms Bðx; yÞ ¼ a À1 ðaðxÞaðyÞ þ aðyÞaðxÞÞ). He also applied this FI to the description of commutativity preserving maps of prime rings (i.e., such maps a that ½x; y ¼ 0 implies ½aðxÞ; aðyÞ ¼ 0). Since then FIs became the fundamental tool in the solution of all remaining unsolved Herstein's problems. [5,6,11,12,15,17,19,26,28,43,44] Besides their role in the study of commutativity preserving maps, [1,7,19] FIs were successfully applied to the study of additive maps maps preserving n th powers, [24] additive maps preserving an arbitrary fixed multilinear polynomial of degree ! 2 [5,13] and the structure theory of certain nonassociative algebras. [10] On the one hand, recently a number of papers on the Lie structure of associative superalgebras [30,31,37,39] have been published. On the other hand, Herstein's problems on Lie maps in associative algebras have been solved. Finally, a question of nonexistence of isomorphisms of certain Lie superalgebras arising from the associative ones has been already posed by Montgomery. [39, p. 571] Therefore it is now natural to consider the analogues of Herstein's problems in the context of associative superalgebras. It has turned out that the current level of the theory of functional identities is not sufficient for solving these problems because the automorphism s defining the Z 2 -graduation on an associative prime superalgebra A is involved in the FIs arising from the study of Lie maps in A. In the case when s is X-inner, i.e., when there exists an invertible element t 2 Q s , the symmetric Martindale ring of A, such that t 2 is central in Q s , these FIs involves the element t. Because of that, the process of exclusion of variables, developed by Beidar in [2] , became essentially more complicated. We overcome this problem by introducing in Sec. 1 the concept of an r-independent set upon which this paper is based. In Sec. 2 we introduce and study the concept of a d-free subset in the context of FIs with d-independent coefficients. This concept is an analog of that of a d-free subset for FIs without coefficients introduced by Beidar and Chebotar in [8] and which, together with corresponding results, [8,9] played the crucial role in the solution of a number of Herstein's Lie map proble...

Functional identities in rings and their applications

“…, y m )]) in two ways. On the one hand, Comparing (14) and (15), we obtain the desired functional identity.…”

“…At the same time some identities of special form were studied with the object of extending the class of rings for which the technique of functional identities can be used ( [11]- [14], [16], [20], [47], [48]). …”

“…For other examples of d-free sets, see the papers [13], [14]. Finally, at the end of the section we discuss generalized functional identities [58] and functional identities of special form ( [47], [48]).…”

Generalized derivations in prime rings

“…Posner Theorem [1] has already been generalized in several derivations [2][3][4][5][6][7] . Bresar [8] proved the form of every additive commutating mapping f on a prime ring.…”