4 - The Structure of Semisimple Rings

Abstract

This chapter discusses the structure of semisimple rings. Any ring that satisfies the ascending chain condition is called Noetherian. A ring satisfying the descending chain condition (d.c.c.) is usually called Artinian. It can be shown that for any commutative ring, the ascending chain condition is equivalent to the existence of a finite basis ring. Any field satisfies trivially both the ascending and descending chain conditions, as it has only the two trivial ideals. It can also be shown that in a ring satisfying the d.c.c. any collection of left ideals contains an ideal that does not properly contain any other ideal, that is, a minimal ideal in the collection. An important structure, from a coding theory point of view, is the group algebra.