Associative algebra

In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. 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 [[algebra over a field|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. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, 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 commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is a ring that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies for all r in R and x, y in the algebra. (This definition implies that the algebra is unital, since rings are supposed to have a multiplicative identity.) Equivalently, an associative algebra A is a ring together with a ring homomorphism from R to the center of A.