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

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