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 Kap-

No abstract

Let R be a ring with unit, and e be an idem potent in R such that (1 - e)Re = 0. In this note we shall explore the relationships between homological properties of R and those of its subring eRe.Examples of such rings are abundant, the most common being perhaps the ring R of all two-by-two upper triangular matrices over a field, where—

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 Kap-