Generalized multiplicative derivations in inverse semirings

On (anti-)multiplicative Generalized Derivations

Let R be a semiprime ring and let F, f : R → R be (not necessarily additive) maps satisfying F (xy) = F (x)y + xf(y) for all x, y ∈ R. Suppose that there are integers m and n such that F (uv) = mF (u)F (v) + nF (v)F (u) for all u, v in some nonzero ideal I of R. Under some mild assumptions on R, we prove that there exists c ∈ C(I) such that c = (m + n)c2, nc[I, I] = 0 and F (x) = cx for all x ∈...

Additively Inverse Semirings

In this paper we show that in a regular additively inverse semiring (S,+, ·) with 1 satisfying the conditions (A) a(a + a′) = a + a′; (B) a(b + b′) = (b + b′)a and (C) a + a(b + b′) = a, for all a, b ∈ S, the sum of two principal left ideals is again a principal left ideal. Also, we decompose S as a direct sum of two mutually inverse ideals.

Semirings whose additive endomorphisms are multiplicative

A ring or an idempotent semiring is associative provided that additive endomorphisms are multiplicative.

Generalized Derivations on Modules

Generalized Derivations on Modules *

Let A be a Banach algebra and M be a Banach left A-module. A linear map δ : M → M is called a generalized derivation if there exists a derivation d : A → A such that δ(ax) = aδ(x) + d(a)x (a ∈ A,x ∈ M). In this paper, we associate a triangular Banach algebra T to Banach A-module M and investigate the relation between generalized derivations on M and derivations on T . In particular, we prove th...