R-algebra

Definition

Let $R$ be a commutative ring. An $R$-algebra is a ring homomorphism $\alpha :R\to S$ such that $\alpha (R)$ is contained in the center of $S$.

That is,

$$\alpha (r)s=s\alpha (r)\forall r\in R,s\in S.$$

As an $R$-algebra

Relationship to other algebras

When $R$ 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	$R$-algebra $\alpha :R\to S$
vector addition	addition in $S$ using ring structure
vector multiplication	multiplication in $S$ using ring structure
scalar multiplication	$\alpha (r)$ the action on $R$