My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
November 19th, 2006, 05:29 PM   #1
Site Founder
 
julien's Avatar
 
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
julien is offline  
 
November 19th, 2006, 05:56 PM   #2
Site Founder
 
julien's Avatar
 
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.
julien is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

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





Copyright © 2019 My Math Forum. All rights reserved.