On modules with finite uniform and Krull dimension (Q1193247)

Let \(R\) be any ring with identity and let \(M\) be a unital right \(R\)- module. By a cyclic subfactor of \(M\) is meant a module of the form \(mR/N\), where \(m\in M\) and \(N\) is a submodule of \(mR\). The module \(M\) is called a \(CS\)-module if every submodule is essential in a direct summand. \textit{B. L. Osofsky} and the reviewer proved that a cyclic module \(M\) has finite uniform (Goldie) dimension if each cyclic subfactor of \(M\) is \(CS\) [J. Algebra 139, No. 2, 342-354 (1991; Zbl 0737.16001)]. Adapting their argument, the authors improve this result by proving that a cyclic module has finite uniform dimension in case each cyclic subfactor is a direct sum of a \(CS\)-module and a module with finite uniform dimension. Various consequences are given. For example, for any ordinal \(\alpha\geq 0\), a finitely generated module \(M\) has Krull dimension at most \(\alpha\) if and only if every cyclic subfactor is a direct sum of an \(M\)-injective module and a module with Krull dimension at most \(\alpha\), generalizing a result of the first two authors and the reviewer [J. Algebra 132, No. 1, 104-112 (1990; Zbl 0706.16013)]. Again, the module \(M\) is locally Artinian if and only if every subfactor is a direct sum of an \(M\)-injective module and a finitely cogenerated module. Finally, the ring \(R\) has the property that every proper cyclic right \(R\)-module is a direct sum of an injective module and a finitely cogenerated module if and only if \(R\) is either right Artinian or \(R\) is a right Ore domain such that \(R/E\) is an Artinian right \(R\)-module for every non-zero right ideal \(E\) [see the first author, Commun. Algebra 18, No. 3, 607-614 (1990; Zbl 0702.16008)]. (In fact, the authors consider modules in the category \(\sigma[M]\) of modules subgenerated by \(M\), and their theorems are consequently somewhat more general than the versions given above).