Semisimple Associative Rings with

An associative ring D with 1 is called semisimple if the D-module D is completely reducible. 

In this section, we shall combine our results about completely reducible modules obtained in Section 8.3 with those about irreducible modules (and artinian simple rings) obtained in Section 8.4 in order to give a complete description of semisimple rings. 

For the remainder of this section, we shall now fix an associative ring with 1 and call it D. 

Lemma 8.5.1 Let be a set of submodules of the D-module D such that D is the sum of the elements in C. Then there exists a finite subset K of C such that D is the sum of the elements in 1C. 

We are assuming that D is a ring with 1. Since D is the sum of the elements in , there exists a finite subset 1C of and, for each element K in 1C, an element dK in K such that 

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In Section 8.3, we focus on completely reducible modules over associative rings with 1. Section 8.4 deals with irreducible modules over these rings. In Section 8.5, we combine the results obtained in these two sections to obtain the famous (and complete) description of semisimple associative rings with 1 which was first given by Joseph Wedderburn and Emil Artin. 

The following two theorems are the main theorems about semisimple associative rings with 1. They are due to Emil Artin cf. [2]. Less general versions have been given earlier by Joseph Wedderburn cf. [39 Theorem 10] and [39 Theorem 17]. 

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