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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regular modules

Author: J. Zelmanowitz
Journal: Trans. Amer. Math. Soc. 163 (1972), 341-355
MSC: Primary 16.56
MathSciNet review: 0286843
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Abstract: In analogy to the elementwise definition of von Neumann regular rings an $ R$-module $ M$ is called regular if given any element $ m \in M$ there exists $ f \in {\operatorname{Hom} _R}(M,R)$ with $ (mf)m = m$. Other equivalent definitions are possible, and the basic properties of regular modules are developed. These are applied to yield several characterizations of regular self-injective rings. The endomorphism ring $ E(M)$ of a regular module $ _RM$ is examined. It is in general a semiprime ring with a regular center. An immediate consequence of this is the recently observed fact that the endomorphism ring of an ideal of a commutative regular ring is again a commutative regular ring. Certain distinguished subrings of $ E(M)$ are also studied. For example, the ideal of $ E(M)$ consisting of the endomorphisms with finite-dimensional range is a regular ring, and is simple when the socle of $ _RM$ is homogeneous. Finally, the self-injectivity of $ E(M)$ is shown to depend on the quasi-injectivity of $ _RM$.

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Keywords: von Neumann regular ring, regular module, finite-dimensional module, projective module, flat module, endomorphism ring, endomorphisms with finite-dimensional range, quasi-injective module
Article copyright: © Copyright 1972 American Mathematical Society