In Section 5.2 of BA we saw that semisimple Artinian rings can be described quite explicitly as direct products of full matrix rings over skew fields (Wedderburn's theorem). Later in Chapters 7 and 8 we shall see what can be said when the Artinian hypothesis is dropped, but in many cases, such as the study of group algebras in finite characteristic, it is important to find out more about the non-semisimple (but Artinian) case. There is now a substantial theory of such algebras which is still developing, and a full treatment is beyond the framework of this book, but some of the basic properties are described here.
KeywordsLocal Ring Direct Summand Left Ideal Projective Module Projective Cover
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