Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Maximal Ideals and the Correspondence Theorem for Rings

  1. Aug 31, 2016 #1
    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ...

    I need some help with understanding the proof of Proposition 5.9 ... ...


    Proposition 5.9 reads as follows:


    ?temp_hash=a8c6b30ae57a4ffe62159d453fb0011e.png


    In the proof of Proposition 5.9, Rotman writes:

    " ... ... The Correspondence Theorem for Rings shows that ##I## is a maximal ideal if and only if ##R/I## has no ideals other than ##(0)## and ##R/I## itself ... ... "

    My question is: how exactly (in clear and simple terms) does Rotman's statement of the Correspondence Theorem for Rings lead to the statement that "##I## is a maximal ideal if and only if ##R/I## has no ideals other than ##(0)## and ##R/I## itself" ... ...

    Hope that someone can help ...

    Peter

    ============================================================

    The above post refers to Rotman's statement of the Correspondence Theorem for Rings, so I am providing a statement of that theorem and its proof, as follows:


    ?temp_hash=a8c6b30ae57a4ffe62159d453fb0011e.png
     
  2. jcsd
  3. Aug 31, 2016 #2
    Maybe I should not be responding to my own post but I have been reflecting on the question in the above post and now suspect that the answer is quite simple and goes along the lines ... ... as follows:

    ##I## maximal

    ##\Longrightarrow## there are no ideals in ##R## that contain ##I## except ##R## itself ...

    ##\Longrightarrow## there are no ideals in ##R/I## (except ##R/I##) since there exists a bijection between the set of ideals of ##R/I## and the ideals of ##R## containing ##I## ... ...

    BUT ... it seems that the only ideal in ##R/I## is ##R/I## itself ... but how do we explain the existence of (0) ...

    Seems that I still need some help ... ...

    Peter
     
  4. Aug 31, 2016 #3

    fresh_42

    Staff: Mentor

    ... and ##I##.

    ... and ##I/I##

    ... and ##I/I##
    We do by identifying ##I/I = (0)##.

    You could also assume an ideal ##\overline{J}= J+I## in ##R/I##. Since ##\overline{0} = 0+I \in \overline{J}##, this means ##I \subseteq J## which means by maximality of ##I## that ##J \in \{I\,,\,R\}##.
    And this holds the other way round, too.
     
    Last edited: Aug 31, 2016
  5. Aug 31, 2016 #4
    Thanks for the help fresh_42 ...

    Just reflecting on what you have said ...

    Peter
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Maximal Ideals and the Correspondence Theorem for Rings
  1. Ideals in Rings (Replies: 2)

Loading...