On a Functional Equation Related to Two-Sided Centralizers

Abstract

Abstract. The main aim of this manuscript is to prove the following result. Let n > 2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k ∈ {2, n − 1, n}. Let us assume that for the additive mappingis also fulfilled. Then T is a two-sided centralizer.In this paper R will denote an associative ring with center Z(R). For an integer n > 1, a ring R is said to be n-torsion free, if for x ∈ R, nx = 0 implies x = 0. The expression xy − yx will be marked by [x, y]. The ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. We indicate by char(R) the characteristic of a prime ring R. Let X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in

Characterization of Jordan centralizers and Jordan two-sided centralizers on triangular rings without assuming unity

“…Vukman [22] showed that an additive mapping T : R → R, where R is a 2-torsion free semiprime ring, satisfying 2T (x 2 ) = T (x)x + xT (x) for all x ∈ R, is a centralizer. To read more about centralizers, we refer the readers to some recent papers [2,3,6,7,13,14,15,19,20,21,23], where further references can be found. So far, many mathematicians have investigated Jordan centralizers on triangular rings (or algebras) with unity, see, e.g.…”