Algebras Over a Field

Abstract

18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that $(ax)(by)=abxy$ for $\,a,\,b\in F$ and $\,x,\,y\in A.$ If A has an identity element, we denote it by $1_A,$ or simply by $1.$ Identifying $\,a\,$ with $\,a1_A$ for every $\,a\in F,$ we can view F as a subring of A.