The class of pure submodules ($𝒫$) and torsion-free images ($ℛ$) of finite direct sums of submodules of the quotient field of an integral domain were first investigated by M. C. R. Butler for the ring of integers (1965). In this case $𝒫=ℛ$ and short exact sequences of such modules are both prebalanced and precobalanced. This does not hold for integral domains in general. In this paper the notion of precobalanced sequences of modules is further investigated. It is shown that as in the case for abelian groups...

In this paper we characterize all prime and primary submodules of the free $R$-module $R^n$ for a principal ideal domain $R$ and find the minimal primary decomposition of any submodule of $R^n$. In the case $n=2$, we also determine the height of prime submodules.

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.