Extended modules (Q847970)

For local ring \(R\) and \(S\) and a flat local homomorphism \((R,{\mathbf m}) \rightarrow (S,{\mathbf n})\), a finitely generated \(S\)-module \(N\) is extended if there exists an \(R\)-module \(M\) such that \(S \otimes_R M\) is isomorphic to \(N\) as \(S\)-module. In this paper, the authors first show that extended modules are closed on direct sums and direct summands. Then, for ring of dimension 2 and 1, they give necessary and sufficient condition for a module to be extended. For one dimension rings, the extendedness problem they show that reduces to the same problem over Artinian (zero-dimensional) rings. Finally, they explain the situation where every finitely generated module is a direct summand of an extended module and they give some observations for Artinian case.