Lattice modules having small cofinite irreducibles (Q5946261)

The authors continue the study of lattice modules for the following particular class. Let \(L\) be a local Noether lattice with ``maximal'' element \(m\), let \({\mathcal M}\) be a Noetherian \(L\)-module with greatest element \(M\). Then \({\mathcal M}\) is said to have small cofinite irreducibles if for every positive integer \(n\) there exists a meet-irreducible \(Q\in{\mathcal M}\) such that \(Q\leq m^n M\) and \({\mathcal M}/Q\) is finite-dimensional. Several characterizations for such \(L\)-modules \({\mathcal M}\) are given by means of \(m\)-primary elements of \({\mathcal M}\) or the \(m\)-adic topology on \({\mathcal M}\). It is also shown that \({\mathcal M}\) has small cofinite irreducibles if and only if the \(L^*\)-module \({\mathcal M}^*\) has cofinite irreducibles, where \(L^*\) and \({\mathcal M}^*\) were defined by the first two authors in Can. J. Math. 22, 327-331 (1970; Zbl 0197.29004). These results are applied to local Noetherian rings \(R\) with maximal ideal \(m\) and Noetherian \(R\)-modules \(M\), the lattice of ideals of \(R\) and the lattice of all \(R\)-submodules of \(M\).