Talk:Maximal ideal

I redirected this page and merged the old content into ideal_(ring theory). I have checked the links and none seemed to refer to ideal in order theory. MathMartin 11:18, 4 Sep 2004 (UTC)


 * I just noticed this comment of mine after starting a new maximal ideal article with material from ideal (ring theory). I changed my opionion and think the topic deserves its own page now. MathMartin 13:45, 19 February 2006 (UTC)

Corrected that every maximal ideal is prime to include the fact that the ring must be commutative. Xantharius 02:01, 21 December 2006 (UTC)

Is it true that 2Z/4Z is not a field? It seems that it is isomorphic to Z2. —Preceding unsigned comment added by 85.101.99.211 (talk) 08:47, 7 July 2007 (UTC)
 * It's not isomorphic. The quotient 2Z/4Z is the two-element non-unital ring where addition is modulo two, but multiplication is constantly zero.—Emil J. 14:58, 5 March 2010 (UTC)

unital
Is not a unital ring the same as a ring with multiplicative identity? If so, why using two names for the same thing?Please editBogdanno (talk) 02:25, 26 August 2010 (UTC)

Properties
"Krull's theorem (1929): Every ring with a multiplicative identity has a maximal ideal."

and

"Every nonzero ring has a maximal ideal."

seem to be overlapping and are confusing. Citation needed? Or some more comments. (77.80.27.163 (talk) 12:11, 10 May 2011 (UTC))

The quotient of a unital commutative ring by a maximal ideal is a field
The introduction contains this statement:

"Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields."

The fact that the quotient of a commutative ring with unit, by a maximal ideal, is a field seems sufficiently important that it ought to be mentioned again in the body of the article, and perhaps even elaborated on with a proof and some examples. The ring of algebraic integers of a number field is such a ring, and these are widely studied.2601:200:C000:1A0:D02F:D749:A7D7:6091 (talk) 19:22, 21 June 2021 (UTC)