Automorphic-differential identities in rings

Abstract

Let R R be a ring and f 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 f on certain subsets of R R to its behavior on all of R R . For example, we prove (1) if R R is prime with ideal I ≠ 0 I \ne 0 such that f ( I ) = 0 f(I) = 0 , then f ( R ) = 0 f(R) = 0 , (2) if R R is a domain with right ideal λ ≠ 0 \lambda \ne 0 such that f ( λ ) = 0 f(\lambda ) = 0 , then f ( R ) = 0 f(R) = 0 , and (3) if R R is prime and f ( λ n ) = 0 f({\lambda ^n}) = 0 , for λ \lambda a right ideal and n ≥ 1 n \geq 1 , then f ( λ ) = 0 f(\lambda ) = 0 . We also prove some generalizations of these results for semiprime rings and rings with no non-zero nilpotent elements.