Extension of Ideal-Theoretic Properties of a Domain to Submodules of Its Quotient Field

We examine when multiplicative properties of ideals extend to submodules of the quotient field of an integral domain. An integral domain R is stable if each non-zero ideal of R is invertible as an ideal over its ring of endomorphisms. We show that an integral domain R is stable if and only if an analogue of this invertibility property extends to submodules of the quotient field of R. By contrast, the class of integral domains for which every non-zero ideal is locally free over its ring of endomorphisms is shown to properly contain the class of domains R for which each submodule of the quotient field is locally free over its ring of endomorphisms, and we give complete characterizations of both classes of domains.