Historically the first rings to be studied (in the second half of the 19th century) were the rings of integers in algebraic number fields. At about the same time the theory of algebras began to develop; its most important landmarks were the Wedderburn structure theorems for semisimple algebras, and the study of the radical. The theories merged when it was realized that the Wedderburn theorems could be stated more generally for Artinian rings. This is the form in which the results will be presented here, in Section 5.2 and 5.3; the formulation for general rings (Jacobson radical and density theorem) will be reserved for FA. As a preparation for this study we examine the form the tensor product takes for algebras in Section 5.4, and in Section 5.5 we introduce scalar invariants. In Section 5.6 algebras are used to define an important number-theoretic function, the Mobius function.
KeywordsLeft Ideal Division Algebra Homomorphic Image Regular Representation Nilpotent Element
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