We characterize prime submodules of $R\times R$ for a principal ideal domain $R$ and investigate the primary decomposition of any submodule into primary submodules of $R\times R.$.
In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics: (i) The existence of prime submodules in some cases, (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.