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International Journal of Mathematics and Mathematical Sciences
Volume 14, Issue 3, Pages 615-618
http://dx.doi.org/10.1155/S0161171291000844

When in a multiplicative derivation additive?

Department of Mathematics, Faculty of Education, Umm Al-Qura University, Taif, Saudi Arabia

Received 29 March 1990; Revised 19 December 1990

Copyright © 1991 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Our main objective in this note is to prove the following. Suppose R is a ring having an idempotent element e(e0, e1) which satisfies: (M1)   xR=0  implies  x=0.(M2)   eRx=0  implies  x=0  (and hence  Rx=0  implies  x=0).(M3)   exeR(1e)=0  implies  exe=0. If d is any multiplicative derivation of R, then d is additive.