Associative Algebra - Formal Definition

Formal Definition

Let R be a fixed commutative ring. An associative 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 ring multiplication is R-bilinear:

for all r ∈ R and x, y ∈ A. We say A is unital if it contains an element 1 such that

for all x ∈ A. Note that such an element 1 must be unique if it exists at all.

If A itself is commutative (as a ring) then it is called a commutative R-algebra.