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.