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Maximal ideals

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Given a ring
R
, a maximal ideal
𝔐
is a non-trivial ideal that is not contained within any ideal other than the whole ring itself. For example,
⟨7⟩
is a maximal ideal in
, as it is a proper ideal of
not contained within any other proper ideal. Compare
⟨21⟩
, which is not a maximal ideal because it is contained within
⟨7⟩
(as well as within
⟨3⟩
). All maximal ideals are prime ideals,[1] but not all prime ideals are maximal ideals.[2] For example,
⟨0⟩
is a prime ideal in
, but it is not a maximal ideal because it is contained within all other ideals in
.

See also

Notes

  1. This assumes a ring with a multiplicative identity. It is possible to find a fake ring without a multiplicative identity in which there is a maximal ideal that is not prime; see Problem 3.10, p. 33 in Watkins (2007).
  2. John J. Watkins, Topics in Commutative Ring Theory. Princeton & Oxford: Princeton University Press (2007): p. 30, Corollary 3.1.