Extensions of Modules

Abstract

In studying modules, as in studying any algebraic structures, the standard procedure is to look at submodules and associated quotient modules. The extension problem then appears quite naturally : given modules A, B (over a fixed ring Λ) what modules E may be constructed with submodule B and associated quotient module A? The set of equivalence classes of such modules E, written E(A, B), may then be given an abelian group structure in a way first described by Baer [3]. It turns out that this group E(A, B) is naturally isomorphic to a group Ext_Λ (A, B) obtained from A and B by the characteristic, indeed prototypical, methods of homological algebra. To be precise, Ext _Λ (A, B) is the value of the first right derived functor of Hom _Λ (- , B) on the module A, in the sense of Chapter IV.