Algebras generated as commutative algebras.

[Kreizhn]

Homework Statement

Let R be a commutative ring. Prove that a commutative R-algebra S is finitely generated as an algebra over R if and only if it is finitely generated as a commutative algebra over R.

The Attempt at a Solution

I'm really not even sure what this is asking. I know that for an R-algebra S, we can view it as being generated as an R-module and as an R-algebra, and that they do not generally agree. However, what's the difference between being generated as an R-algebra and being generated as a commutative R-algebra?

 

[micromass]

Hi Kreizhn!

I'm not sure, but I'll have a go at it. I think that S is finitely generated as a commutative A-algebra if there is a surjection

$$A[T_1,...,T_n]\rightarrow S$$

However, S is finitely generated as an A-algebra if there is a surjection

$$A<T_1,...,T_n>\rightarrow S$$

Where $A<T_1,...,T_n>$ are polynomials in n variables, but the variables don't commute. So we don't have $T_1T_2=T_2T_1$.

What you need to show is that if S is commutative, then these two notions are equivalent.

 

[Kreizhn]

Okay thanks. I'll give it a shot and see what I can come up with.