On finite-dimensional torsion-free modules and rings

Abstract

1. This note is concerned primarily with a study of modules having zero singular submodule, called torsion-free modules, over finitedimensional rings. In some of our results we do not require that the ring be finite-dimensional but only that direct sums of torsion-free injective modules be injective, a property shown by Mark Teply to be equivalent to the ring containing no infinite direct sum of torsionfree left ideals. As might be expected, torsion-free modules over these latter rings behave very much like the usual torsion-free modules over (commutative) integral domains. For example, we show in Theorem 1 that direct sums of torsion-free injectives are injective if and only if every torsion-free module contains a unique maximal injective submodule. Also, in analogy to the case for Abelian groups, we establish that a torsion-free module over a finite-dimensional torsion-free ring contains a homomorphic image of every nonzero torsion-free module if and only if it contains a faithful injective submodule (Theorem 3). This enables us to show in Theorem 4 that a semiprime finite-dimensional torsion-free ring has a projective injective envelope if and only if it is semisimple Artinian. A result for torsion-free prime rings without any restricted chain conditions similar to Theorem 3 is also obtained. Finally, over self-injective finitedimensional rings we show that all torsion-free modules are injective and completely reducible (Corollary 6). We assume throughout that R is a ring with identity and all R-modules are unitary left R-modules.