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Algebra-Modules generalize Ring-Modules?

  1. Jan 11, 2012 #1
    Throughout this post, I consider "R-module" and "representation of R" to be the same.
    Similarly, I consider "A-module" and "representation of A" to be the same.

    I define of a representation of a unital ring R as an Abelian group M together with a homomorphism from R to the unital ring of endomorphisms of M.

    I define of a representation of a unital associative algebra A as a vector space V together with a homomorphism from A to the unital associative algebra of endomorphisms of V.

    Question:
    Is it so that the notion of a representation of a unital associative algebra generalises the notion of a representation of a unital ring? I think I've heard this result, but can't find it by a Google search.

    It would seem strange if it were true, since the notion of a unital ring actually generalizes the notion of a unital associative algebra.
     
  2. jcsd
  3. Jan 11, 2012 #2

    morphism

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    An algebra is a type of ring and an algebra endomorphism is in particular an endomorphism of the underlying ring. Similarly, a vector space is in particular an abelian group and an endomorphism of a vector space is an endomorphism of the underlying abelian group. So if you start with a rep V of an algebra A, i.e. with a map A->EndV, then you can forget the algebra structure on A and the vector space structure on V, i.e. you can view A as just a ring and V as just an abelian group, and everything will still work out. The map A->EndV is still a homomorphism of rings; the only new thing here is that we're taking End in the category of abelian groups.

    Does this clarify things for you? Apologies if I misunderstood what you were asking.
     
  4. Jan 14, 2012 #3
    Thank you. What you say seems to confirm my intuition and suggests that perhaps I was wrong. What I thought I had heard somewhere was that a representation of a group is a special case of a representation of an algebra.

    I think I may have found the key to the problem. It says here under "Associative Algebras", that the notion of an associative algebra is a generalization of the notion of a ring. Maybe that's the key here.
     
    Last edited: Jan 14, 2012
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