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.
Read more about this topic: Associative Algebra
Famous quotes containing the words formal and/or definition:
“The bed is now as public as the dinner table and governed by the same rules of formal confrontation.”
—Angela Carter (1940–1992)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)