Algebra ring theory: Encyclopedia II - Algebra ring theory - Formal definition

Algebra ring theory - Formal definition

Let R be a commutative ring. An R-algebra is a set A which has the structure of both a ring and an R-module in such a way that ring multiplication is an R-bilinear map. Explicity, we must have

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

Starting with an R-module A, we get an R-algebra by equipping A with an R-bilinear map A × A → A such that

for all x, y, and z in A. This R-bilinear map then gives A the structure of a ring.

Conversely, starting with a ring A, we get an R-algebra by providing a ring homomorphism whose image lies in the center of A. The algebra A can then be thought of as an R-module by defining

for all r ∈ R and x ∈ A.

Adapted from the Wikipedia article "Formal definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki