The Maximal ideal reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Maximal ideal

In mathematics the term ideal refers to two different concepts: either to ideals in ring theory or to ideals in order theory. However, in both cases ideals are certain sets which can be partially ordered via subset inclusion. Consequently, a maximal ideal is a maximal element in the corresponding orders of proper ideals (i.e. ideals other than the whole ring/poset). Alternatively, maximal ideals are directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.

For ideals in ring theory further information is found in the article on prime ideals. Ideals of order theory, including maximal ideals, are treated in ideal (order theory).

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