On extensions of modules

In this paper we study closely Yonedaʹs correspondence between short exact sequences and the Ext1 group. We prove a main theorem which gives conditions on the splitting of a short exact sequence after taking the tensor product with R/I, for any ideal I of R. As an application, we prove a generalization of Miyataʹs theorem on the splitting of short exact sequences and we improve a proposition of Yoshino about efficient systems of parameters. We introduce the notion of sparse module and we show that is a sparse module provided that there are finitely many isomorphism classes of maximal Cohen–Macaulay modules having multiplicity the sum of the multiplicities of M and N. We prove that sparse modules are Artinian. We also give some information on the structure of certain Ext1 modules.