On torsion free distributive modules (Q839695)

Let \(R\) be a commutative ring with identity and \(M\) be an \(R\)-module. Then \(M\) is said to be distributive if, \(K\cap(L+N)=K\cap L+K\cap N\), for all submodules \(K, L\) and \(N\) of \(M\). In this paper a torsion free distributive module is characterized in terms of its primal submodules and submodules of irreducible components. The main result of the paper asserts that a torsion free \(R\)-module \(M\) is distributive if and only if any primal submodule of \(M\) is irreducible and equivalently each submodule of \(M\) can be represented as an intersection of irreducible isolated components. (Reviewer's comment: In the proof of Theorem 3.6 of the part \((ii)\rightarrow (i)\), the argument \(N_{m}/(mN)_{m}\) is a finite dimensional vector space over the field \(R_{m}/(mR)_{m}\) implies that \(N_{m}/(mN)_{m}\) is of dimension one may not be correct unless \(N_{m}/(mN)_{m}\) is indecomposable and there is no justification given as to why this is the case. The same argument is also used to prove Theorem 3.7).