Equivalence and Duality for Module Categories

Abstract

So far our emphasis has been on studying rings in terms of the module categories they admit—that is, in terms of the representations of the rings as endomorphism rings of abelian groups. As we shall see the Wedderburn Theorem for simple artinian rings can be interpreted as asserting that a ring R is simple artinian if and only if the category _R M is “the same” as the category _D M for some division ring D. On the other hand, if D is a division ring, then the theory of duality from elementary linear algebra asserts that the categories _D FM and FM _D of finitely generated left D-vector spaces and right D-vector spaces are “duals” of one another.