On $h$-local integral domains

Abstract

Related to the question of determining the integral domains with the property that finitely generated modules are a direct sum of cyclic submodules is the question of determining when an integral domain is $h$-local, especially for Bezout domains. Presented are ten equivalent conditions for a Prüfer domain with two maximal ideals not to be $h$-local. If $R$ is an integral domain with quotient field $Q$, if every maximal ideal of $R$ is not contained in the union of the rest of the maximal ideals of $R$, and if $Q/R$ is an injective $R$-module, then $R$ is $h$-local; and if in addition $R$ is a Bezout domain, then every finitely generated $R$-module is a direct sum of cyclic submodules. In particular if $R$ is a semilocal Prüfer domain with $Q/R$ an injective $R$-module, then every finitely generated $R$-module is a direct sum of cyclic submodules.