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Identity in a subring

  1. Mar 9, 2008 #1

    quasar987

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    [SOLVED] Identity in a subring

    1. The problem statement, all variables and given/known data
    In Dummit & Foote on the section on tensor product of modules (10.4 pp.359), the authors write

    "Suppose that the ring R is a subring of the ring S. Throughout this section, we always assume that [tex]1_R=1_S[/tex] (this ensures that S is a unital R-module)."

    Now, I just want to make sure I'm not missing something. Can R be a subring with identity whose identity 1_R is different from 1_S???

    I would say "no" because S, together with its multiplication operation, forms a monoid... and in a monoid M, the identity e is the only element with the ability to do me=em=m for any and all m in M. So if 1_R were an identity for R different than 1_S, it would mean 1_R*r = 1_S*r = r and we would have two different element with the ability to act on the elements of r like an identity, which contradicts the fact that S is a (multiplicative) monoid.

    So instead, perhaps a less confusing way to write the above quoted passage would be to say,

    "Suppose that the ring R is a subring of the ring S. Throughout this section, we always assume that R contains [tex]1_S[/tex] (this ensures that S is a unital R-module)."

    Yes? Thanks for the feedback.
     
  2. jcsd
  3. Mar 9, 2008 #2

    HallsofIvy

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    You are clearly assuming that any ring has a multiplicative identity. Is Dummit and Foote assuming that?
     
  4. Mar 9, 2008 #3

    Dick

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    Take S to be the ring of 2x2 matrices and R to be the subring consisting consisting of matrices [[x,0],[0,0]] for any x. [[1,0],[0,0]] is a unit for R, but not for S.
     
    Last edited: Mar 9, 2008
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