The Laskerian condition for nonfinitely generated modules
Let \(R\) be a commutative ring and let \(M\) be an \(R\)-module. \(M\) is said to have property \((L)\) if every submodule of \(M\) is an intersection of finitely many submodules \(V_ i\) of \(R\) such that each \(M/V_ i\) is primary. Finitely generated modules with this property are called Laskerian in the literature. The main result in the first section of this paper is that if \(R\) is noetherian, then \(M\) has property \((L)\) if and only if it is a sum \(A+B\) of submodules, with \(A\) finitely generated, and \(R/\text{Ann}_ R(B)\) artinian. -- The second section gives various characterizations of modules that have a primary decomposition, again in the case of a noetherian base ring. -- The final section of the paper provides examples, with particular attention to the case where \(R\) is of dimension one. Modules that have both primary and coprimary decompositions are also characterized.
scientific article; zbMATH DE number 118333