Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication. In this article associative algebras are assumed to have a multiplicative unit, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra. Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies for all r ∈ R and x, y ∈ A. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that for all x ∈ A. Note that such an element 1 must be unique. In other words, A is an R-module together with (1) an R-bilinear map A × A → A, called the multiplication, and (2) the multiplicative identity, such that the multiplication is associative: for all x, y, and z in A. (Technical note: the multiplicative identity is a datum, while associativity is a property. By the uniqueness of the multiplicative identity, 'unitarity' is often treated like a property.) If one drops the requirement for the associativity, then one obtains a non-associative algebra.