Simple algebras

Abstract

This Chapter will be purely algebraic in nature; this means that we will operate over a groundfield, subject to no restriction except commutativity, and carrying no additional structure. All fields are understood to be commutative. All algebras are understood to have a unit, to be of finite dimension over their ground-field, and to be central over that field (an algebra A over K is called central if K is its center). If A, B are algebras over K with these properties, so is A⊗ _K B; if A is an algebra over K with these properties, and L is a field containing K, then A _L = A⊗ _K L is an algebra over L with the same properties. Tensor-products will be understood to be taken over the groundfield ; thus we write A⊗B instead of A⊗ _K B when A, B are algebras over K, and A⊗L or A _L , instead of A⊗ _K L, when A is an algebra over K and L a field containing K, A _L being always considered as an algebra over L.