# Associative Algebra 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`*. The term *K-algebra* is used to mean an *associative algebra over the field `K`*. The standard example of a K-algebra is a ring of square matrices over a field `K`, with the usual matrix multiplication. Associative algebras are assumed to have a *multiplicative identity*, denoted by `1` (they are sometimes called *unital associative algebras*), and that all *rings*, as well as all *ring homomorphisms*, are *unital* . A **commutative algebra** is an associative algebra that has a *commutative multiplication*, or, equivalently, an associative algebra that is also a *commutative ring*.