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 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.

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