Results 1 to 2 of 2

Math Help - R-module homomorphism surjectivity

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    23

    R-module homomorphism surjectivity

    Let I be a nilpotent ideal in a commutative ring R, let M and N be R-modules and let \phi : M \rightarrow N be an R-module homomorphism. Show that if the induced map \bar{\phi} : \frac{M}{IM} \rightarrow \frac{N}{IN} is surjective, then \phi is surjective.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by zelda2139 View Post
    Let I be a nilpotent ideal in a commutative ring R, let M and N be R-modules and let \phi : M \rightarrow N be an R-module homomorphism. Show that if the induced map \bar{\phi} : \frac{M}{IM} \rightarrow \frac{N}{IN} is surjective, then \phi is surjective.
    so I^k=0, for some k \geq 1. since \bar{\phi} is surjective, we have: \frac{N}{IN}=\bar{\phi}\left(\frac{M}{IM}\right)=\  frac{\phi(M)+IN}{IN}. thus: N=\phi(M)+IN, which gives us: N=\phi(M)+I(\phi(M)+IN)=\phi(M)+I^2N, because

    I\phi(M) \subseteq \phi(M). repeating this we'll get: N=\phi(M)+I^2(\phi(M) + IN)=\phi(M) + I^3N = \cdots = \phi(M) + I^kN=\phi(M). \ \Box
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Unique R-Module Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 2nd 2011, 09:17 PM
  2. Z-module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 28th 2009, 03:55 AM
  3. simple question on R module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 23rd 2009, 07:12 AM
  4. projective, module homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 13th 2009, 05:57 PM
  5. Surjectivity of a natural ring homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 24th 2008, 08:04 AM

Search Tags


/mathhelpforum @mathhelpforum