Results 1 to 2 of 2

Math Help - ring morphism and modules

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    48

    ring morphism and modules

    Let A and B rings, M a B-module and \phi: A \rightarrow B a ring morphism. Prove that the action a._{\phi}x=\phi(a)x defines a structure of A-module over M.

    thanks!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by roporte View Post
    Let A and B rings, M a B-module and \phi: A \rightarrow B a ring morphism. Prove that the action a._{\phi}x=\phi(a)x defines a structure of A-module over M.

    thanks!!
    the action is clearly well-defined because if a = a' and x = x', then \phi(a)x=\phi(a')x=\phi(a')x'. now M is a B-module and \phi is a homomorphism. therefore:

    (a + b)\cdot x = \phi(a+b)x=(\phi(a) + \phi(b))x=\phi(a)x + \phi(b)x=a \cdot x + b \cdot x. similarly: a \cdot (x+y)=\phi(a)(x+y)=\phi(a)x + \phi(a)y=a \cdot x + a \cdot y. finally:

    (ab) \cdot x = \phi(ab)x=(\phi(a)\phi(b))x=\phi(a)(\phi(b)x)=a \cdot (b \cdot x). \ \ \Box
    Last edited by NonCommAlg; November 18th 2008 at 08:11 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prove the Artinian ring R is a division ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 8th 2011, 03:53 AM
  2. Prove f is a group morphism and establish 1 condition
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 19th 2011, 05:55 AM
  3. example of prime ring and semiprime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 27th 2011, 05:23 PM
  4. Frobenius Morphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 14th 2011, 06:28 PM
  5. morphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 22nd 2009, 11:44 PM

Search Tags


/mathhelpforum @mathhelpforum