engleski

On (anti-)multiplicative generalized derivations

Let $R$ be a semiprime ring and let $F, f : R \to R$ be (not necessarily additive) maps satisfying $F(xy)=F(x)y+xf(y)$ for all $x, y \in R.$ 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 $c \in C(I^{; ; \perp \perp}; ; )$ such that $c=(m+n)c^2, $ $nc[I^{; ; \perp \perp}; ; , I^{; ; \perp \perp}; ; ]=0$ and $F(x)=cx$ for all $x \in I^{; ; \perp \perp}; ; .$ The main result is then applied to the case when $F$ is multiplicative or anti-multiplicative on $I.$

Additivity ; ring ; semiprime ring ; prime ring ; derivation ; generalized derivation ; homomorphism ; anti-homomorphism

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