On the quotients of countable direct products of modules modulo direct sums
In this paper \(R\) is assumed to be a commutative domain with 1, and \(R \neq Q\), for the field of quotients \(Q\). Inspired by a known result of Hulanicki and Mycielski, the authors look into the module \(M^* = \prod M_ n/ \bigoplus M_ n\), for countable sets \(\{M_ 1, \dots, M_ n, \dots\}\) of \(R\)-modules. \(M^ 1 = \bigcap_{0 \neq r \in R} rM\) denotes the first Ulm submodule of \(M\). The culminating result of the paper is as follows: If p.d. \(Q = 1\), and \(M_ n\) are reduced torsion modules, then \(M^*\) satisfies: (a) \(M^{*1}\) is a divisible \(R\)-module; and (b) \(M^*/M^{*1}\) is isomorphic to a submodule of elements of countable support in a product of \(R\)-complete modules. (c) \(M^*/M^{*1}\) is \(R\)-complete, for every choice of torsion \(R\)-modules \(M_ n\), if and only if \(Q/R\) is a countably generated \(R\)-module.
scientific article; zbMATH DE number 638255
scientific article