Finite Dimensional Algebras

Abstract

Let k be a discrete field. A k-algebra is a ring A that is also a vector space over k, satisfying λ(ab) = (λa)b = a(λb) for each λ in k and a, b in A. If A and B are k-algebras, then a homomorphism from A to B is a ring homomorphism that is also a k-linear transformation. The term finite dimensional, when applied to a structure S that is a vector space over k, like a k-algebra, signifies that S is a finite-dimensional vector space over k.