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 fiat.) 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.
Preliminaries.Throughout this paper, unless otherwise indicated, all modules over a ring R will be understood to be left A-modules. R will always have a unit, and every module will be unitary. All homomorphisms of A-modules will be written on the right so that if M is an A-module and S'=HomB (M, M) then M becomes an R -S bimodule, sometimes written RMS. For all notions of homological algebra the reader is referred to [3]. This section consists of definitions, notation terminology, and basic facts about projective modules which will be used in the later ones.Let A be a ring, M an A-module, and N a submodule of M. We say N is small in M if whenever A is a submodule of M with N+K=M then K=M. Dually, N is large in M if N n A=0 always implies A=0.
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