November 19th, 2006, 05:29 PM  #1 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Maximal Ideal
>if a is a noninvertible element of a ring then it >belongs to a maximal ideal 
November 19th, 2006, 05:56 PM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7 
I can consider the ring R to be unitary (since we are talking about invertibility) and commutative (easier for the notations, and we don't lose anything important). By the axiom of choice (or more precisely by the maximum principle), you can find a maximal (for the inclusion) sequence (a) C I1 C...C In C...C... such that the In are all strict ideals of R, and (a) is the ideal generated by a. The union UIk is also an ideal of R containing a, and it is strict, because it does not contain 1 (otherwise one of the Ik contains 1, meaning Ik=R, which is absurd). Whence UIk is a maximal ideal of R. 

Tags 
ideal, maximal 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Maximal ideal  limes5  Abstract Algebra  7  January 6th, 2014 03:43 PM 
maximal ideal  cummings123  Abstract Algebra  1  February 27th, 2013 05:06 AM 
maximal ideal of the ring Q trivial question??  rayman  Abstract Algebra  2  November 21st, 2012 03:16 AM 
generating a maximal ideal, is this correct?  rayman  Abstract Algebra  3  October 6th, 2012 09:17 AM 
ideal and maximal  silvi  Abstract Algebra  2  January 20th, 2010 03:05 AM 