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Transactions of the American Mathematical Society

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



Endomorphism rings of projective modules

Author: Roger Ware
Journal: Trans. Amer. Math. Soc. 155 (1971), 233-256
MSC: Primary 16.40
MathSciNet review: 0274511
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Abstract: The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect.

Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is a regular ring. It is shown that many other equivalent ``regularity'' conditions characterize regular modules. (For example, every homomorphic image is flat.) Every projective module over a regular ring is regular and a number of examples of regular modules over nonregular rings are given. A structure theorem is obtained: every regular module is isomorphic to a direct sum of principal left ideals. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. Conversely, over a commutative ring a projective module having a regular endomorphism ring is a regular module. Examples are produced to show that these results are the best possible in the sense that the hypotheses of finite generation and commutativity are needed. An application of these investigations is that a ring $ R$ is semisimple with minimum condition if and only if the ring of infinite row matrices over $ R$ is a regular ring.

Next projective modules having local, semiperfect and left perfect endomorphism rings are studied. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring.

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Keywords: Projective module, endomorphism ring, Jacobson radical, von Neumann regular ring, local ring, semiperfect ring, left perfect ring, left $ T$-nilpotent, flat module, semiprime ring, row finite matrices, injective module
Article copyright: © Copyright 1971 American Mathematical Society