Semi-irreducible Zariski spaces of modules (Q2878789)

The authors study strongly irreducible and semi-irreducible modules over a commutative ring, and semi-irreducible Zariski spaces of modules. All the needed definitions are formulated in the paper, including a review of Zariski spaces (Chapter 2). Let \(R\) be a commutative ring, and let \(M\) an \(R\)-module. Let \(K\) be a submodule of \(M\). \(K\) is called \textit{strongly irreducible}, if for every two submodules of \(N,N^{\prime}\) of \(M\) such that \(N\cap N^{\prime}\subseteq K\), either \(N\) or \(N^{\prime}\) is contained in \(K\). By Lemma 3.1 in the paper, a submodule \(K\) of \(M\) is \textit{semi-irreducible} if and only if for every \(m\in M\) and \(r\in R\), the inclusion \(rM \cap Rm\subseteq K\) implies that either \(m\in K\) or \(r \in (K: M)\). The authors prove that \(M\) satisfies the colon property on strongly irreducible submodules if this property is satisfied locally, that is, by the \(R_{\mathfrak p}\)-module \(M_{\mathfrak p}\) for every prime ideal \(\mathfrak p\) of \(R\). They ask whether a module satisfying the colon property locally on semi-irreducible submodules, necessarily satisfies this property, the converse being true. The authors characterize semi-irreducible and strongly irreducible submodules of a finitely generated module over a Dedekind domain. They also show that two finitely generated modules over a Dedekind domain with isomorphic semi-irreducible Zariski spaces share certain common invariants.