Some results concerning additive mappings and derivations on semiprime rings

Abstract

We consider an optimization problem given by a discrete inclusion, whose trajectories are constrained to closed sets. Necessary optimality conditions in the form of the maximum principle and in terms of local generalized derived cones of constraints are obtained.

Identities With Additive Mappings in Semiprime Rings

“…In [18] Vukman and the first named author generalized the result proved by Brešar and Hvala for prime rings [9].…”

“…Furthermore, Brešar [7] proved that every additive commuting mapping of a prime ring R is of the form x → λx + ζ(x), where λ is an element of the extended centroid C and ζ : R → C is an additive mapping. For results concerning commuting mappings, centralizing mappings and related problems we refer the reader to [1,[5][6][7][8][9][10][11][12][13]18,[22][23][24][25][26][27][28] where further references can be found.…”

Remarks on Certain Identities with Derivations on Semiprime Rings

“…Thus, let R be a 2-torsion free semiprime ring and let f : R ! R be an additive mapping such that [f (x), x 2 ] = 0 holds for all x 2 R. Then f must be commuting on R. This result was proved by Vukman and Fošner in [9]. Moreover, the same conclusion is true if f satisfies the relation [f (x), x n ] = 0, x 2 R, where n is a fixed positive integer and R is an n!-torsion-free semiprime ring (see [8], Theorem 2).…”

“…The aim of the present paper is to generalize the results obtained in [9]. We first fix some notation.…”

“…In the present paper, we continue a series of papers dealing with arbitrary additive maps of prime and semiprime rings satisfying certain identities (see [1][2][3][4][5]9] and the references therein). In particular, we generalize the main results obtained in [9].…”