Derivations, generalized derivations, and *-derivations of period 2 in rings

Abstract

The aim of this article is to discuss the existence of certain kinds of derivations and *-derivations that are of period 2. Moreover, we obtain the form of generalized reverse derivations and generalized left derivations of period 2 .

Generalized Jordan triple derivations associated with Hochschild 2-cocycles in semiprime rings

“…A ring R is m-torsion free, where m = 0 is an integer if whenever ml = 0 with l ∈ R, then l = 0. An R-bimodule W is a left and right R-module such that l(wq) = (lw)q for all w ∈ W and l, q ∈ R. A bi-additive map γ: R × R −→ W is called a Hochschild 2-cocycle, if lγ(q, t) − γ(lq, t) + γ(l, qt) − γ(l, q)t = 0 for all l, q, t ∈ R, and γ is called symmetric if γ(l, q) = γ(q, l) for all l, q ∈ R. Nakajima [10] has introduced the notions of generalized derivations and generalized Jordan derivations associated with Hochschild 2-cocycles. An additive map G: R −→ W is called a generalized derivation associated with a Hochschild 2-cocycle γ: R × R −→ W if G(lq) = G(l)q + lG(q) + γ(l, q) for all l, q ∈ R, and G is called a generalized Jordan derivation associated with Hochschild 2-cocycle γ if G(l 2 ) = G(l)l + lG(l) + γ(l, l) for all l ∈ R. If γ = 0, then G means the usual derivation and Jordan derivation.…”