Titlebook: Classically Semisimple Rings; A Perspective Throug Martin Mathieu Textbook 2022 The Editor(s) (if applicable) and The Author(s), under excl

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The Correspondence of William Burnside,Finiteness conditions on the ideal structure of a ring allow us to develop a more detailed structure theory. In the noncommutative framework, it is more natural to do this for (one-sided) modules rather than for (one-sided) ideals, so we shall follow this approach right from the beginning.

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,The Misnamings of Playfair’s Axiom,In this chapter, which is at the very heart of our exposition, we shall apply module theory to study the structure of a certain class of rings; these will be termed (.) .. To this end, we will, once again, introduce some special classes of modules and investigate their properties.

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Noetherian Modules,Finiteness conditions on the ideal structure of a ring allow us to develop a more detailed structure theory. In the noncommutative framework, it is more natural to do this for (one-sided) modules rather than for (one-sided) ideals, so we shall follow this approach right from the beginning.

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Artinian Modules,The dual condition to “ACC” is “DCC”, the descending chain condition which we shall discuss in the present chapter. The resulting modules are termed . after Emil Artin (1898–1962).

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Simple and Semisimple Modules,In this chapter, which is at the very heart of our exposition, we shall apply module theory to study the structure of a certain class of rings; these will be termed (.) .. To this end, we will, once again, introduce some special classes of modules and investigate their properties.

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,L’équivalence duale de catégories: ,?,t. Its main benefit lies in the fact that it allows us to convert bilinear mappings into homomorphisms of abelian groups. The relations between tensor products and homomorphism groups is fundamental and will lead us to the concept of adjoint functor in the later part of the chapter.

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,Analysis and Synthesis in Robert Simson’s ,g .[.] is semisimple, provided . is a finite group. For any field ., the elements of . form a basis of the .-vector space .[.] and if the ring .[.] is semisimple, then it is necessarily Artinian, hence finite dimensional (Corollary . and Exercise .). As a result, we cannot expect .[.] to be semisimple for an infinite group ..