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?