The Morita Theorem

Abstract

In Sect. 2.3 we have noted that modules over a division algebra D and modules over the simple algebra M _n (D) are “equally structured”. Results of Sect. 2.6 show that, in general, modules over isotypic semisimple algebras possess the same properties: such modules have isomorphic endomorphism rings, etc. In Sect. 3.5 these results have been extended to projective modules over arbitrary isotypic algebras (Lemma 3.5.5). It turns out that one can remove the requirement of projectivity: All modules over isotypic algebras are equally structured. However, in order to formulate this statement properly, it is necessary to introduce a number of concepts which presently play an important role in various areas of mathematics. Above all, it is the concept of a category and a functor, as well as the notion of an equivalence of categories, which appears to be a mathematical formulation of the expression “equally structured”.