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:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (1564–1616)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)