Slender rings and modules

Abstract

An abelian group B is called slender if every homomorphism from ^ * nto B * s continuous for the discrete topology on B and the topology that JJf Z has as a cartesian product if Z (the additive group of integers) is considered discrete. It can be seen that if B is slender, then every homomorphism from ^ * nto & * s continuous for the same topologies, where A< are arbitrary abelian groups. Slender groups were completely characterized by Nunke, and it follows from his characterization that all countable reduced torsion free groups are slender. Nunke's results depend on the fact that Z is slender (proved by Specker) and the use of some homological machinery. In this paper, slender modules are studied over an arbitrary ring. A ring R will be called slender if R is a slender i?-module. In Theorem 1, a generalization of the Baire Category Theorem is used to show that a countable module over an arbitrary ring is slender if vxB = 0, where m ranges over the set of ideals in R which are not zero divisors for B. It follows that countable torsion free reduced modules over any (countable) integral domain are slender. In Theorem 2, it is shown that a commutative ring R is slender if there exists an infinite set of maximal ideals with the property that the intersection of any infinite subset is 0.