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Ideals

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Given a ring , an ideal is a non-empty subset of that is itself a group under addition, and, for any and it is the case that .[1]

In order for to be a group under addition, it must contain 0, and it must be closed under addition, meaning that for any it follows that also.

For example, in , , which consists of all numbers in of the form is an ideal.

See also: prime ideals, principal ideals, maximal ideals.

  1. John J. Watkins, Topics in Commutative Ring Theory. Princeton & Oxford: Princeton University Press (2007): p. 11, Definition 2.1.