Wedderburn’s Theorem shows that the class of semisimple algebras is very limited. On the other hand, Proposition 3.1a suggests that Artinian algebras are semisimple “up to a radical.” In fact, this is the case. All that is missing from a proof is the result that rad A A is an ideal. We will establish this fact in Section 4.1. The rest of the chapter is concerned with properties and characterizations of the radical, a theorem about nilpotent algebras, and the radicals of group algebras.
KeywordsPrime Ideal Left Ideal Group Algebra Division Algebra Nilpotent Element
Unable to display preview. Download preview PDF.