Delta operation on modules, prime and radical submodules and primary decomposition

Abstract

Let R be a commutative ring with identity and M be an R-module. In this paper, in order to study prime submodules, radical submodules and primary decompositions in finitely generated free R-modules, we introduce and study an operation ∆: (formula presented) defined by (formula presented). In particular, using this operation we give a characterization of prime submodules of (formula presented), in terms of prime submodules of M. As an application, we present a characterization of prime submodules of finitely generated free modules. Also we present a formula for the prime radical of submodules of (formula presented). Moreover, we state some conditions under which primary decompositions of submodules of M lift to (formula presented)