n-Commuting skew higher derivation on semiprime rings

Centralizers on semiprime rings

The main result: Let R be a 2-torsion free semiprime ring and let T : R → R be an additive mapping. Suppose that T (xyx) = xT (y)x holds for all x, y ∈ R. In this case T is a centralizer.

Generalized Jordan Triple Higher ∗−Derivations on Semiprime Rings

Let R be an associative ring not necessarily with identity element. For any x, y ∈ R. Recall that R is prime if xRy = 0 implies x = 0 or y = 0, and is semiprime if xRx = 0 implies x = 0. Given an integer n ≥ 2, R is said to be n−torsion free if for x ∈ R, nx = 0 implies x = 0.An additive mapping d : R → R is called a derivation if d(xy) = d(x)y + yd(x) holds for all x, y ∈ R, and it is called a...

Centralizers on prime and semiprime rings

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R. Suppose that [S(x), T (x)]S(x) + S(x)[S(x), T (x)] = 0 is fulfilled for all x ∈ R. If S 6= 0 (T 6= 0) then there exists λ from the extende...

A note on derivations in semiprime rings

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

Dependent Elements of Derivations on Semiprime Rings