Direct products of modules

Abstract

Introduction.It is a well-known and basic result of homological algebra that the direct product of an arbitrary family of injective modules over any ring is again injective [3, p. 8].Such is not the case for projective modules, as is evidenced, for example, by a result of Baer [7, p. 48] which states that the direct product of a countably infinite number of copies of the ring of rational integers is not a free abelian group.It is thus natural to ask for the precise ideal-theoretic conditions which are forced upon a ring by the requirement that its projective modules be preserved by direct products in the manner just described.In this paper we shall present a solution to this problem, as well as an answer to the corresponding question for flat modules.We then exhibit several applications of these results.First, we derive information concerning semi-hereditary rings which, when applied to integral domains, yields immediately characterizations of Prüfer rings due to Hattori [5] (see also [6]).The second application also concerns integral domains.Let us call a torsion module over an integral domain £ a UT-module if it is a direct summand of every £-module of which it is the torsion submodule.We prove that, if every £-module of bounded order is a UT-module, then £ must be a Dedekind ring (the converse is well-known; see [8, p. 334]).As a final application, we obtain a partial solution to the following question of Köthe [9]: For which rings £ is it true that every left £-module is a direct sum of cyclic modules?We prove that, if £ has the weaker property that every left £-module is a direct sum of finitely generated modules, then £ satisfies the minimum condition on left ideals and every indecomposable injective left £-module has finite length.This generalizes a result of Kaplansky and Cohen [4].Our investigations along these lines were motivated to some extent by the interesting observation of Bass [2] that left Noetherian rings are characterized by the property that their injective left modules are preserved by direct sums.We have, with his kind permission, included this in our paper, as we shall need it in the proof of another result.