Separability conditions for vector R-modules

Abstract

One of the recurring themes in module-theory is the problem of describing classes of modules which are obtained as the closure with respect to direct sums or products of a well-understood class A of modules. For instance, the classes of completely decomposable groups and vector groups arise as closures of the class A _ℤ of subgroups of the rationals with respect to direct sums and direct products, respectively. These two classes of groups are also closed with respect to direct summands by the Baer-Kulikov-Kaplansky Theorem [8, Theorem 86.7] and O’Neill’s theorem [12]. In contrast, the closure of an arbitrary class A under direct sums or direct products may not have this closure property as many examples demonstrate (see [8] for details). This and similar difficulties can, in many cases, be avoided by restricting the discussion to families A which are semi-rigid, i.e. which have the property that any two modules A, B ∈ A with Rom _R (A, B) ≠ 0 and Hom _R (B, A) ≠ 0 are isomorphic. Arnold, Hunter, and Richman showed in [6] that the class of finitely A-decomposable modules is closed with respect to direct summands if A is semi-rigid. Here, an -R-module M is finitely A-decomposable if there are A ₁,..., A _n ∈ A with the property that $M \cong { \oplus _{i\mathop = \limits^n 1}}{P_i}$for suitable A _i -projective modules P _i of finite A _i -rank whenever i = 1,..., n. Similarly, an R-module V is an A-vector-module if it is isomorphic to Π _{i ∈I} A _i where each A _i ∈ A.