Results 1 to 2 of 2

Thread: Ring

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    1

    Ring

    Let R be a ring, and I be an ideal of G.
    Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
    Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

    can someone help me please
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by evonnova View Post
    Let R be a ring, and I be an ideal of G.
    Show that there is a one to one and onto correspondence between the subrings of R/I and those subrings of R which contain I.
    Hint: maybe the "natural" homomorphism f: R-> R/I defined by f(x)= x+I

    can someone help me please
    Saying "one-to-one and onto correspondence" is not necessary because a one-to-one correspondence is defined as a one-to-one and onto.

    Define $\displaystyle \pi: R\mapsto R/I$ to be the natural projection. If $\displaystyle S$ is a subring of $\displaystyle R$ then $\displaystyle \pi (S)$ is a subring of $\displaystyle R/I$. If $\displaystyle K$ is a subring of $\displaystyle R/I$ then $\displaystyle \pi^{-1} (K)$ is a subring of $\displaystyle R$ which contains $\displaystyle R$.

    Let $\displaystyle X$ be the set of all subrings of $\displaystyle R$ which contain $\displaystyle I$ and $\displaystyle Y$ be the set of all subrings of $\displaystyle R/I$. By above paragraph define $\displaystyle \hat \pi: X\mapsto Y$ as $\displaystyle \hat \pi (x) = \pi (x)$ for all $\displaystyle x\in X$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 23rd 2011, 06:36 AM
  2. [SOLVED] Prove the Artinian ring R is a division ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jun 8th 2011, 03:53 AM
  3. example of prime ring and semiprime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 27th 2011, 05:23 PM
  4. Ideals of ring and isomorphic ring :)
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 24th 2009, 03:23 AM
  5. ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Jul 27th 2009, 09:07 PM

Search Tags


/mathhelpforum @mathhelpforum