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 formal Washington dinner party has all the spontaneity of a Japanese imperial funeral.”
—Simon Hoggart (b. 1946)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)