A question about commutative $R$-algebras.

Question

Aluffi describes a commutative $R$-algebra as a ring homomorphism $\alpha:R\to S$ where $S$ is commutative.

Going by this definition, how is $R[x_1,x_2,\dots,x_n]$ a commutative $R$-algebra? What is $S$ here?

Answer 1

The $R$-algebra is $R \to R[x_1,\dotsc,x_n]$, $r \mapsto r$ (viewed as a constant polynomial).

Answer 2

You call a ring $S$ an $R$-algebra, if you have such a ring homomorphism $R \to S$ (more strictly speaking, the definition that I know is: an $R$-algebra is a commutative ring $S$ together with a ring homomorphism $R\to S$). Often such a homomorphism happens to be canonical or obvious, hence one omits stating the homomorphism.

In your case one says $S=R[x_1,\cdots,x_n]$ is an $R$-algebra, by the natural embedding $R \to R[x_1,\cdots,x_n]$.

Hope that helps.