(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

3. 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?

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# Algebras generated as commutative algebras.

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