On degenerations of modules

Abstract

We generalize a result of Zwara concerning the degeneration of modules over Artinian algebras to that over general algebras. In fact, let R be any algebra over a field and let M and N be finitely generated left R-modules. Then, we show that M degenerates to N if and only if there is a short exact sequence of finitely generated left R-modules 0→Z → φ ψ M⊕Z→N→0 such that the endomorphism ψ on Z is nilpotent. We give several applications of this theorem to commutative ring theory.