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Homework Help: Ring and Linear Transformation

  1. Mar 8, 2006 #1
    Let [tex]g(x)\in F[x][/tex], [tex]T\in L(V)[/tex]. Let [tex]F[T][/tex] be a ring generated by [tex]g(T)[/tex]. Show that if [tex]g(T)[/tex] is invertible, then [tex]g^{-1}(T)\in F[T][/tex].

    No idea what do do.
     
  2. jcsd
  3. Mar 8, 2006 #2

    Hurkyl

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    The statement of the problem is confusing. I'm going to assume V is a vector space over the field F.

    Now, F[T] is not just "a" ring -- it is a very specific ring. It is (isomorphic to) the subring of L(V) that is generated by F and T.

    So, I'm confused when you say that you let it be "a ring generated by g(T)".

    Did you mean to say something to the effect if:

    "Suppose g(T) generates F[T]"

    or maybe

    "Let F[g(T)] be the ring generated by g(T)"

    or even just

    "So g(T) is an element of F[T]"

    ?


    (I suspect you meant the first one -- work out what that really means)
     
  4. Mar 8, 2006 #3
    It's either the first or third. Let's assume the first, since I must show that the inverse of g(T) is a positive powered polynomial.
     
  5. Mar 8, 2006 #4

    Hurkyl

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    Okay, so what does it mean that g(T) generates F[T]? Any interesting particular cases?
     
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