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Centralizer near-rings that are rings

Part of: Modules, bimodules and ideals Generalizations Rings and algebras arising under various constructions

Published online by Cambridge University Press:  09 April 2009

Jutta Hausen  and
Johnny A. Johnson
Jutta Hausen
Affiliation:
University of Houston, Houston, Texas 77204-3476, e-mail addresses: [email protected], [email protected]
Johnny A. Johnson
Affiliation:
University of Houston, Houston, Texas 77204-3476, e-mail addresses: [email protected], [email protected]
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Abstract

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Given an R-module M, the centralizer near-ring ℳR (M) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳR(M) is not a ring but is a near-ring containing the endomorphism ring ER(M) of M. Necessary and/or sufficient conditions are derived for ℳR(M) to be a ring. For the case that R is a Dedekind domain, the R-modules M are characterized for which (i) ℳR(M) is a ring; and (ii)ℳR(M) = ER(M). It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16S50: Endomorphism rings; matrix rings 16Y30: Near-rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 2 , October 1995 , pp. 173 - 183
Copyright
Copyright © Australian Mathematical Society 1995

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