Automorphic-differential identities in rings

Abstract:

Let $R$ be a ring and $f$ an endomorphism obtained from sums and compositions of left multiplications, right multiplications, automorphisms, and derivations. We prove several results relating the behavior of $f$ on certain subsets of $R$ to its behavior on all of $R$. For example, we prove (1) if $R$ is prime with ideal $I \ne 0$ such that $f(I) = 0$, then $f(R) = 0$, (2) if $R$ is a domain with right ideal $\lambda \ne 0$ such that $f(\lambda ) = 0$, then $f(R) = 0$, and (3) if $R$ is prime and $f({\lambda ^n}) = 0$, for $\lambda$ a right ideal and $n \geq 1$, then $f(\lambda ) = 0$. We also prove some generalizations of these results for semiprime rings and rings with no non-zero nilpotent elements.