Results 1 to 2 of 2

Thread: Projective Modules

  1. #1
    Junior Member
    Joined
    Jun 2011
    Posts
    34

    Projective Modules

    Prove that I_6 is simultaneously an injective and a projective module over itself.
    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
    22

    Re: Projective Modules

    Quote Originally Posted by jcir2826 View Post
    Prove that I_6 is simultaneously an injective and a projective module over itself.
    As a $\displaystyle \mathbb{Z}_6$-module, $\displaystyle \mathbb{Z}_6$ is free. Surely you know that all free modules are simultaneously projective. Thus, all we have to show is that $\displaystyle \mathbb{Z}_6$ is injective. To do this we can use Baer's criterion, which says that $\displaystyle \mathbb{Z}_6$ will be injective over $\displaystyle \mathbb{Z}_6$ if and only if we can extend every $\displaystyle \mathbb{Z}_6$-homomorphism $\displaystyle \mathfrak{a}\to\mathbb{Z}_6$ can be extended to $\displaystyle \mathbb{Z}_6\to\mathbb{Z}_6$ for every $\displaystyle \mathfrak{a}$ a left ideal of $\displaystyle \mathbb{Z}_6$. So...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Idempotents and Projective R-Modules
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 16th 2013, 08:48 AM
  2. Replies: 0
    Last Post: Jan 12th 2012, 02:23 AM
  3. modules that are both injective projective
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Jan 6th 2012, 12:38 AM
  4. Replies: 1
    Last Post: Dec 5th 2011, 12:17 PM
  5. map between projective modules over a Dedekind domain
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Apr 14th 2011, 11:26 PM

Search Tags


/mathhelpforum @mathhelpforum