Associative Algebra - Associativity and The Multiplication Mapping

Associativity and The Multiplication Mapping

Associativity was defined above quantifying over all elements of A. It is possible to define associativity in a way that does not explicitly refer to elements. An algebra is defined as a vector space A with a bilinear map

(the multiplication map). An associative algebra is an algebra where the map M has the property

Here, the symbol refers to function composition, and Id : A → A is the identity map on A.

To see the equivalence of the definitions, we need only understand that each side of the above equation is a function that takes three arguments. For example, the left-hand side acts as

Similarly, a unital associative algebra can be defined as a vector space A endowed with a map M as above and, additionally, a linear map

(the unit map) which has the properties

Here, the unit map η takes an element k in K to the element k1 in A, where 1 is the unit element of A. The map t is just plain-old scalar multiplication: ; the map s is similar: .