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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Automorphic-differential identities in rings

Author: Jeffrey Bergen
Journal: Proc. Amer. Math. Soc. 106 (1989), 297-305
MSC: Primary 16A72; Secondary 16A12
MathSciNet review: 967482
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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.

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Article copyright: © Copyright 1989 American Mathematical Society

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