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.
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