A general method is given for
constructing countably generated modules with a number of bizarre properties
over any commutative Artinian ring which is not a principal ideal ring. The
main result shows that if R is a commutative local Artinian ring which is
not a principal ideal ring, and the residue class field of R is k, then any
pathological property that holds for some k[t]-module also holds for a suitable
R-module. This method gives easy and uniform proofs of many known results
(and some new ones) concerning modules over these rings. A theorem of
A. L. S Corner’s, concerning countable endomorphism rings of torsion-free
Abelian groups, is generalized to algebras over suitable discrete valuation rings,
and applied to obtain further pathological results for modules over Artinian
rings.
