Regular and semisimple modules.

is a T-ring.(1)—>(3b). A semisimple module is torsion.(3)—>(1). This is an immediate consequence of Lemma 4.Neither (3a) nor (3b) is sufficient by itself to imply (1). TomShorem has shown (unpublished) that the product of an infinitenumber of fields (or an infinite number of copies of a single field)is a T-ring. Thus (3a) does not imply (1). The subring