Abstract
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.
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© 1982 Springer-Verlag New York Inc.
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Pierce, R.S. (1982). The Radical. In: Associative Algebras. Graduate Texts in Mathematics, vol 88. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-0163-0_4
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DOI: https://doi.org/10.1007/978-1-4757-0163-0_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-0165-4
Online ISBN: 978-1-4757-0163-0
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