Kostial I. Beidar

On functional identities and d-free subsets of rings. II

We show that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions)provided that the ring does not satisfy a standard identity of low degree. Obtained results have applications to the study of Lie maps of prime rings (Lie ideals of prime rings and skew elements of prime rings with involution)and to the study of Lie-admissible algebras and Lie homomorphisms of Lie algebras of Poisson algebras.

On functional identities and commuting additive mappings

We study functional identities of degree m. As an application of obtained results, we describe additive mappings fi:A→Aof a prime ring A satisfying a functional identity of the form Provided that Ais not algebraic of bounded degree ≤nover the extended centroid. A number of results on k-commuting additive mappings and commuting traces of m-additive mappings, obtained earlier, are special cases of our ones.

On Herstein’s Lie map conjectures, I

First published in Transactions- American Mathematical Society in Vol.353, No.10, pp.4235-4260, published by the American Mathematical Society

Jordan isomorphisms of triangular matrix algebras over a connected commutative ring

Abstract Let C be a 2-torsionfree commutative ring with identity 1, and let T r ( C ) , r⩾2 , be the algebra of all upper triangular r×r ( r⩾2 ) matrices over C . Then C contains no idempotents except 0 and 1 if and only if every Jordan isomorphism of T r ( C ) onto an arbitrary algebra over C is either an isomorphism or an anti-isomorphism.

On Lie derivations of Lie ideals of prime algebras

AbstractLetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureAc and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let

ON SURJECTIVE LIE HOMOMORPHISMS ONTO LIE IDEALS OF PRIME RINGS

\bar {\mathcal{R}} = {\mathcal{R}}/{\mathcal{Z}}\left( {\mathcal{R}} \right)

Extended Jacobson density theorem for rings with derivations and automorphisms

be the factor Lie algebra and let δ: