ON PRIME SUBMODULES AND PRIMARY DECOMPOSITIONS IN TWO-GENERATED FREE MODULES

Abstract

In this paper, we consider the free R-module R⊕R, where R is an arbitrary commutative ring with identity. We give a full characterization for prime submodules of R⊕R and a useful primeness test for a finitely generated submodule of R⊕R. We study the existence of primary decomposition of a submodule of R⊕R and characterize the minimal primary decomposition. As applications of our results, we give some examples of primary decompositions in R⊕R.