An abelian group B is called
slender if every homomorphism from ∏1∞Z into B is continuous for the
discrete topology on B and the topology that ∏1∞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 ∏1∞Ai into B is
continuous for the same topologies, where Ai 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 R-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 ∩mB = 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.
In §2, reflexive modules over a slender ring are studied, and it is shown that a
projective module is reflexive if it is generated by a set with cardinality smaller than
the first measurable cardinal. In §3, Nunke’s characterization is extended to modules
over a slender Dedekind domain having only countably many ideals. The general
approach in §3 follows Nunke’s, but all homological machinery is avoided by the use
of topological techniques.
