Commutative rings over which every module has a maximal submodule

Abstract

In this note we characterize those commutative rings over which every nonzero module has a maximal submodule. Professor Hyman Bass in [1, p. 470] states the following conjecture: a ring R is left perfect if, and only if, every nonzero left R-module has a maximal submodule, and R has no infinite set of orthogonal idempotents. For commutative rings we also show that Bass' conjecture is true. Throughout, R will be a ring with identity, J will denote the Jacobson radical of R, and S will denote the ring R/J. We use the word module to mean unital module. If M is a left R-module, rad M denotes the radical of M, that is, the intersection of the maximal submodules of M. If m GM, then R(O: m) = {rER: rm = O} . A left ideal L in R is left T-nilpotent if for each sequence { ri} 1 in L, there is some positive integer k with r1 rk =0. A submodule B of an R-module M is small in M if B + M' = M where M' is a submodule of M implies that M' = M. The first lemma occurs as a remark in [1, p. 470].