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Algebras generated as commutative algebras.

  1. Jun 21, 2011 #1
    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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  3. Jun 21, 2011 #2

    micromass

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    Hi Kreizhn! :smile:

    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

    [tex]A[T_1,...,T_n]\rightarrow S[/tex]

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

    [tex]A<T_1,...,T_n>\rightarrow S[/tex]

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

    What you need to show is that if S is commutative, then these two notions are equivalent.
     
  4. Jun 21, 2011 #3
    Okay thanks. I'll give it a shot and see what I can come up with.
     
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