Matrix Rings, Categories of Modules, and Morita Theory
This last chapter offers an introduction to the basic categorical aspects of the theory of rings and modules. Since its introduction in the 1940s by Eilenberg and MacLane, the categorical viewpoint has been widely accepted by working mathematicians. For ring theorists especially, the convenient use of the categorical language in dealing with modules serves to provide a unifying force for the subject, and has subsequently become an indispensable tool in its modern study. In this chapter, we shall focus on two of the most important concepts in the application of category theory to rings and modules, namely, the equivalence and duality between two categories of modules. Both of these concepts come from the ground-breaking paper of K. Morita , which set in place the basic treatment of these topics pretty much as they are in use today.
KeywordsDivision Ring Endomorphism Ring Matrix Ring Artinian Ring Category Equivalence
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