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 /¡-local, especially for Bezout domains. Presented are ten equivalent conditions for a Prüfer domain with two maximal ideals not to be /i-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 Rmodule is a direct sum of cyclic submodules.In [8] E. Matlis defines an integral domain P to be «-local if every nonzero ideal of P is contained in only a finite number of maximal ideals and if every nonzero prime ideal of R is contained in only one maximal ideal. It is an open question whether an integral domain is «-local if it has the property that all finitely generated modules are a direct sum of cyclic submodules. If we had an affirmative answer to this question, then using the results of [1] we would have a generalized fundamental theorem of Abelian groups, i.e., we would have a characterization of the integral domains with the property that every finitely generated module is a direct sum of cyclic submodules.All rings will be commutative with identity and all modules will be unitary modules. R will always denote a ring and £2 will denote the set of maximal ideals of P. If A is a module, then .4* will denote the nonzero elements of A. If P happens to be an integral domain, then Q will denote the field of fractions of R and K will denote the P-module Q/R. If A is an P-module, the P-topology on A is the topology with the submodules rA, r G R*, being a subbase for the open neighborhoods of 0 in A. If R is an integral domain, then H will denote the completion of R in the P-topology. H is a ring and H = proj limr£Ä» R/Rr =
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.