Results 1 to 2 of 2

Math Help - Composition of Ring Homomorphisms

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Composition of Ring Homomorphisms

    Let f:R\to S, g:S\to T be functions. Let g\circ f: R\to T be the composition of f and g. Show that if f and g are homomorphisms, then so is g\circ f.

    Let u,v\in R.
    (g\circ f)(uv)=g(f(uv))=g(f(u)f(v))=g(f(u))g(f(v))
    =(g\circ f)(u)(g\circ f)(v)

    (g\circ f)(u+v)=g(f(u+v))=g(f(u)+f(v))=g(f(u))+g(f(v))
    =(g\circ f)(u)+(g\circ f)(v)

    (g\circ f)(1)=g(f(1))=g(1)=1

    Therefore, g\circ f is a homomorphism. Correct?
    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

    Re: Composition of Ring Homomorphisms

    Quote Originally Posted by dwsmith View Post
    Let f:R\to S, g:S\to T be functions. Let g\circ f: R\to T be the composition of f and g. Show that if f and g are homomorphisms, then so is g\circ f.

    Let u,v\in R.
    (g\circ f)(uv)=g(f(uv))=g(f(u)f(v))=g(f(u))g(f(v))
    =(g\circ f)(u)(g\circ f)(v)

    (g\circ f)(u+v)=g(f(u+v))=g(f(u)+f(v))=g(f(u))+g(f(v))
    =(g\circ f)(u)+(g\circ f)(v)

    (g\circ f)(1)=g(f(1))=g(1)=1

    Therefore, g\circ f is a homomorphism. Correct?
    Right.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: September 7th 2011, 04:52 AM
  2. Ring homomorphisms - statement
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 12th 2011, 01:18 PM
  3. What kind of proof to use? Ring Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 30th 2010, 06:47 PM
  4. Ring Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 18th 2010, 03:51 PM
  5. Ring-Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 24th 2008, 06:31 AM

/mathhelpforum @mathhelpforum