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 Kap-

n-Coherence Relative to a Hereditary Torsion Theory

“…Note that ϕ 1 is an isomorphism, so, by the Five Lemma, we have that ϕ 2 is also an isomorphism. So, N is τ-FP by Proposition 2.5 in [4], and it shows that A is τ-2-FP.…”

“…(1)⇒(3) In case, n � 0, then the result holds by Lemma 3.1 in [4]. In case, n � 1, then there is an exact sequence of right R-modules 0 ⟶ K ⟶ F ⟶ A ⟶ 0, where F is finitely generated free and K is τ-FP.…”

“…Let τ � (T, F) be a hereditary torsion theory. en, by eorem 3.3 in [4], it is easy to see that R is right τ-coherent if and only if every finitely presented right R-module is τ-2-FP.…”

“…Use induction on n. If n � 0, then the result holds by Proposition 2.5(3) in [4]. Assume that the result holds when n � k. en, when n � k + 1, suppose A is a τ-(k + 2)-presented module.…”

“…We recall also that a right R-module M is called FP-injective [2] or absolutely pure [3] if Ext 1 R (A, M) � 0 for every finitely presented right R-module A; a left R-module M is flat if and only if Tor R 1 (A, M) � 0 for every finitely presented right R-module A; a ring R is right coherent [4] if every finitely generated right ideal of R is finitely presented, or equivalently, if every finitely generated submodule of a projective right R-module is finitely presented. FP-injective modules, flat modules, coherent rings, and their generalizations have been studied extensively by many authors.…”