Definition

Let be a commutative ring. An -algebra is a ring homomorphism such that is contained in the center of .

That is,

As an -algebra

When is a field this notion corresponds to an associative algebra, which can be thought of as a vector space with a vector multiplication operation.

Why are these two the same?

Associative algebra	-algebra
vector addition	addition in using ring structure
vector multiplication	multiplication in using ring structure
scalar multiplication	the action on