CHAPTER 4 - Wedderburn Structure Theorems

Abstract

In case of algebras, there are many ways to form a new algebra from one or more given algebras, and to decompose a given algebra into some component algebras. The component algebras might be of simpler structure that facilitates the study of the given algebra. Because algebra is a ring, the general ideas in connection with forming the direct sum or the tensor product of some given rings are applicable to the case of algebras. The general ideas in connection with decomposing a given ring into a direct sum of some of its ideals or decomposing a ring into a tensor product of some of its sub-rings are applicable to the case of algebras. An algebra is nilpotent if it has no non-zero idempotent element. An algebra is nilpotent if it is a nil algebra. This chapter proves some theorems concerning simple and semi-simple algebras. Every semi-simple and simple algebra has a unit element. A semi-simple algebra is irreducible if it is simple. A semi-simple algebra with no non-trivial ideal is a simple algebra.