Results 1 to 3 of 3

Math Help - Universal arrow from an abelian group to the forgetful functor U: Rng --> Ab

  1. #1
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1

    Universal arrow from an abelian group to the forgetful functor U: Rng --> Ab

    This is one of the exercises in Maclane's book.

    Find, for any given abelian group G, a universal arrow from G to the forgetful functor U:\mbox{Rng}\to\mbox{Ab} which forgets the multiplicative structure.
    I can describe the following construction, which I think fits the bill.

    My first guess is that R_G should be a quotient of a suitably defined free object on the elements of G. Take the free monoid M on the elements of G, and then take the monoid ring \mathbb{Z}[M]=F on the elements of M. Then, let I be the ideal of F generated by elements of the form \tilde a+ \tilde b- \tilde c, where \tilde a, \tilde b, \tilde c are the canonical images of a, b, c \in G via the obvious "inclusion" G \to F, with a+b-c=0 in G. Then, if I am not mistaken, R_G=F/I is the desired "universal ring" on the abelian group G.

    Does this seem correct? If we have an morphism of abelian groups h:G\to U(S), we can extend h to a morphism of monoids M \to S by defining h(a_1\dots a_k)=h(a_1)\dots h(a_k) for any word a_1\dots a_k in M (and by definition of the empty product, the identity of M is mapped to the identity of S). Then we can extend this map to a morphism of rings \mathbb{Z}[M]\to S by linearity. But there is no guarantee at this point that the additive structure imposed by G will be respected, so we have to quotient out by the ideal described above. The resulting map \tilde h seems to do the job.

    In Maclane's book, the category \mbox{Rng} is meant to represent the category of rings, with arrows the ring morphisms preserving identity elements. However, I can read on Wikipedia that this category is now called \mbox{Ring}, and that \mbox{Rng} denotes the category of pseudo-rings. This distinction seems reasonable. If this is the case, we should replace \mbox{Rng} by \mbox{Ring} above. I think that the construction of a "universal" pseudo-ring on G would be the same as above, except that we would take the free semigroup on the elements of G instead of the free monoid.

    If you have any comments, or insight, or if you can suggest a simplification of the above construction, I would be glad to hear it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Bruno J. View Post
    This is one of the exercises in Maclane's book.

    I can describe the following construction, which I think fits the bill.

    My first guess is that R_G should be a quotient of a suitably defined free object on the elements of G. Take the free monoid M on the elements of G, and then take the monoid ring \mathbb{Z}[M]=F on the elements of M. Then, let I be the ideal of F generated by elements of the form \tilde a+ \tilde b- \tilde c, where \tilde a, \tilde b, \tilde c are the canonical images of a, b, c \in G via the obvious "inclusion" G \to F, with a+b-c=0 in G. Then, if I am not mistaken, R_G=F/I is the desired "universal ring" on the abelian group G.

    Does this seem correct? If we have an morphism of abelian groups h:G\to U(S), we can extend h to a morphism of monoids M \to S by defining h(a_1\dots a_k)=h(a_1)\dots h(a_k) for any word a_1\dots a_k in M (and by definition of the empty product, the identity of M is mapped to the identity of S). Then we can extend this map to a morphism of rings \mathbb{Z}[M]\to S by linearity. But there is no guarantee at this point that the additive structure imposed by G will be respected, so we have to quotient out by the ideal described above. The resulting map \tilde h seems to do the job.

    In Maclane's book, the category \mbox{Rng} is meant to represent the category of rings, with arrows the ring morphisms preserving identity elements. However, I can read on Wikipedia that this category is now called \mbox{Ring}, and that \mbox{Rng} denotes the category of pseudo-rings. This distinction seems reasonable. If this is the case, we should replace \mbox{Rng} by \mbox{Ring} above. I think that the construction of a "universal" pseudo-ring on G would be the same as above, except that we would take the free semigroup on the elements of G instead of the free monoid.

    If you have any comments, or insight, or if you can suggest a simplification of the above construction, I would be glad to hear it.
    That looks good to me. I mean, you kind of just did what you "should" do. That said, I myself have only recently started studying \textbf{Cat}-theory myself.

    P.S. Does Mclane define a forgetful functor to be anything that "forgets"? I am using Adamek et. al and they reserve it merely for the obvious functors F:\textbf{C}\to\textbf{Set}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Yes, the notion of "forgetful functor" is wider than, say, the notion of a faithful set-valued functor. You'll find a good description on Wikipedia.

    Thank for your input. It seems good to me also; I did just what I "should" do, perhaps, but it still took me all day to come up with!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Forgetful Functor
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: September 8th 2011, 04:23 PM
  2. Is the subgroup of an abelian group always abelian?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 6th 2009, 11:38 PM
  3. Abelian group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 20th 2009, 10:26 PM
  4. Abelian Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 16th 2009, 07:14 PM
  5. abelian group
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: May 26th 2009, 11:59 AM

Search Tags


/mathhelpforum @mathhelpforum