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 ∆ : (M ⊕ R) 2 → M defined by ∆(m + r, m + r) = r m − rm. In particular, using this operation we give a characterization of prime submodules of M ⊕ R, 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 M ⊕ R. Moreover, we state some conditions under which primary decompositions of submodules of M lift to M ⊕ R.