Projective modules

Abstract

Abstract.In this note we prove that if R is a ring satisfying a polynomial identity and P is a projective left .R-module such that P is finitely generated modulo the Jacobson radical, then P is finitely generated. As a corollary we get that if R is a ring still satisfying a polynomial identity and M is a finitely generated flat /R-module such that M/JM is ^//-projective, then M is Rprojective, J denotes the Jacobson radical. 0. Introduction. In this note R denotes an associative ring with an identity element, J denotes the Jacobson radical of R and all modules considered are unitary fi-modules.In [6] D. Lazard proved that a projective module P over a commutative ring R, such that P/JP is finitely generated, is finitely generated. Furthermore he remarked that is was unknown whether or not the commutativity of R was essential for the validity of the result. In this note we prove that the theorem holds in any P.L ring.1. Flat and projective modules. Let us consider the following properties of a ring R.(i) Any projective left fi-module P, such that P/JP is finitely generated, is finitely generated.(ii) Any finitely generated submodule of a projective module is contained in a maximal submodule.(iii) Any finitely generated flat left fi-module M, such that M/JM is R/Jprojective, is projective.First we prove that if (i) holds for all projective left fi-modules, then (iii) will also hold. This result might be well known, but I have not been able to find it in the literature. Proposition 1.1. If(i) holds in the ring R, then (iii) will also hold.Proof. Suppose Af is a finitely generated flat left fi-module, such that M/JM is fi/Z-projective, and M is not projective. We have an exact sequence

When every finitely generated flat module is projective

“…Here we have only partial results, based on the following Fact 5.4 [8,Lemma 2.4]. Let P be a projective right module over an arbitrary ring R. If P /P Jac(R) is finitely generated and so is P /P I for every prime ideal I , then P is finitely generated.…”

A Variation of Coretractable Modules

“…(b) In [14,Theorem 2.5], it is shown that any ring with polynomial identity is an L-ring. (c) [25,Corollary 3.3] shows that if a ring R has either left or right Krull dimension, then R is an L-ring.…”

Supplement submodules and a generalization of projective modules

“…As we have mentioned in the introduction, rings for which every supplement submodule of a finitely generated projective module is a direct summand have been widely studied (see [4,[6][7][8][11][12][13]16]). The aim of this section is to extend some of the characterizations of these rings to those in which every supplement submodule of a (non-necessarily finitely generated) projective module is a direct summand.…”

“…It is known that rings with polynomial identity, semihereditary rings and rings whose prime quotients rings are left Goldie rings verify the condition (i) of Corollary 3.4 for finitely generated projective modules (see [7,11] for more examples). Moreover, left noetherian rings and commutative domains verify the conditions of Corollary 3.4.…”

“…In general, supplement submodules are strict generalizations of direct summands. Those rings for which every supplement submodule of a finitely generated projective module is a direct summand have been widely treated in the literature (see [6][7][8]11,13,16]). However, it is an open problem to give an internal characterization of these rings (see [9]).…”