A note on generalized (m, n)-Jordan centralizers

Abstract

Abstract. The aim of this paper is to define generalized (m, n)-Jordan centralizers and to prove that on a prime ring with nonzero center and char(R) = 6mn(m+n)(m+2n) every generalized (m, n)-Jordan centralizer is a two-sided centralizer.Throughout, R will represent an associative ring with a center Z(R). Let n ≥ 2 be an integer. A ring R is said to be n-torsion free if for x ∈ R, nx = 0 implies x = 0. Recall that R is prime if aRb = {0} implies a = 0 oris fulfilled for all x ∈ R. One can easily prove that every derivation is a Jordan derivation, but converse is in general not true. A classical result due to Herstein [5, Theorem 3.3] asserts that a Jordan derivation on a prime ring of characteristic different from two is a derivation. A brief proof of Herstein's result can be found in [3]. This result was extended to 2-torsion free semiprime rings by Cusack [4] (see [2] for an alternative proof).An additive mapping T : R → R is called a left (right) centralizer if T (xy) = T (x)y (T (xy) = xT (y)) holds for all pairs x, y ∈ R. If R has the identity element, then T : R → R is a left centralizer if and only if T is of the form T (x) = ax for all x ∈ R, where a ∈ R is a fixed element. For a semiprime ring R, left centralizers are of the form T (x) = qx for all x ∈ R, where q is a fixed element of a Martindale right ring of quotients Q r (see, for example, Chapter 2 in [1]). An additive mapping T : R → R is called a left (right) Jordan centralizer if T (x 2 ) = T (x)x (T (x 2 ) = xT (x)) holds for all x ∈ R. We call an additive mapping T : R → R a two-sided centralizer 2010 Mathematics Subject Classification: 16N60, 39B05.

More on the Generalized (m,n)-Jordan Derivations and Centralizers on Certain Semiprime Rings

“…Inspired by the work of Vukman [15,19], Fošner [8] introduced more generalized version of (m, n)-Jordan centralizers as follows: Let m, n ≥ 0 be two fixed integers with m + n = 0. An additive mapping T : R −→ R is called a generalized (m, n)-Jordan centralizer if there exists an (m, n)-Jordan centralizer T 0 :…”

“…Inspired by the work of Vukman [15,19], Fošner [8] introduced more generalized version of (m, n)-Jordan centralizers as follows: Let m, n ≥ 0 be two fixed integers with m + n = 0. An additive mapping…”

“…[8], Conjecture 1) Let m, n ≥ 1 be two fixed integers, let R be a semiprime ring with suitable torsion restrictions, and let T : R −→ R be a generalized (m, n)-Jordan centralizer. Then T is a two-sided centralizer.…”

“…[8], Lemma 1) Let m, n ≥ 0 be distinct integers with m + n = 0, let R be a ring, and let T : R −→ R be a nonzero generalized (m, n)-Jordan centralizer with an associated (m, n)-Jordan centralizer T 0 . Then, 2(m+n) 2 T (xyx) = mnT (x)xy + m(2m + n)T (x)yx − mnT (y)x 2 + 2mnxT 0 (y)x − mnx 2 T 0 (y) + n(m + 2n)xyT 0 (x) + mnyxT 0 (x) for all x, y ∈ R.…”

The Characterization of Generalized Jordan Centralizers on Triangular Algebras

“…Lately several authors investigated ( , )− Jordan centralizers or their related mappings on rings and algebras. Some of the results can be found in ( [5,[12][13][14][15][16][17][18][19][20]).…”