Abstract
A multiplicative left centralizer for an associative ring R is a map satisfying T(xy) = T\(x)y for all x,y in R. T is not assumed to be additive. In this paper we deal with the additivity of the multiplicative left centralizers in a ring which contains an idempotent element. Specially, we study additivity for multiplicative left centralizers in prime and semiprime rings which contain an idempotent element.
Highlights
He called the maps of the form x → ax + xb where a, b are fixed elements in R “inner generalized derivations"
In parallel to the works of Martindale [7] and Daif [3], we ask the following question for a multiplicative generalized derivation: When is a multiplicative generalized derivation additive, that is when a multiplicative generalized derivation is a generalized derivation? Under some conditions, we give an answer for this question, as a consequence of the result we obtain for left centralizers
When the ring R has an identity element, it is easy to prove that any multiplicative generalized derivation and any multiplicative left centralizer is additive as follows
Summary
He called the maps of the form x → ax + xb where a, b are fixed elements in R “inner generalized derivations". When the ring R has an identity element, it is easy to prove that any multiplicative generalized derivation and any multiplicative left centralizer is additive as follows. The same conclusion holds in the case R is a commutative prime (or semiprime) ring without an identity element.
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