Vol. 60, No. 2, 1975

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Countably generated modules over commutative Artinian rings

Robert Breckenridge Warfield, Jr.

Vol. 60 (1975), No. 2, 289–302

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.

Mathematical Subject Classification 2000
Primary: 13C05
Received: 30 May 1974
Published: 1 October 1975
Robert Breckenridge Warfield, Jr.