Algebra (ring theory)

As algebra over a commutative ring or R algebra (wherein R is a commutative ring ) is defined as an algebraic structure, which consists of a module to a commutative ring and an additional, compatible with the module structure ( algebraic ) multiplication. In particular, an algebra over a commutative ring is a generalization of algebra over a field.

General definition

Let R be a commutative ring, A an R- module and

A binary operation on A called " multiplication ".

The pair is called " R- algebra " if the multiplication is bilinear, ie for arbitrary elements and ring elements applies:

First, there is neither associative nor commutative nor the existence of the identity of the algebra multiplication required.

Special definition

Let R be a commutative ring. An R - algebra is a tuple. Where A is a commutative unitary ring and a ring homomorphism.

Properties

• R thus defined algebra A can be regarded as a module by virtue of R
• An R - algebra A is called finite if it is construed as an R- module is finitely generated
• An R - algebra A is called finitely generated if there is a surjective algebra homomorphism for a

Algebra homomorphism

An R - algebra homomorphism f from to is a ring homomorphism from A to B for the rule is that

Examples

• Each ring is a -algebra is an algebra over the commutative ring of integers.
• Every commutative ring is an algebra over itself
• The polynomial ring over a ring is a finitely generated, but not finite algebra.