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Given a ring
, a
maximal ideal is a non-trivial
ideal that is not contained within any ideal other than the whole ring itself. For example,
is a maximal ideal in
, as it is a proper ideal of
not contained within any other proper ideal. Compare
, which is not a maximal ideal because it is contained within
(as well as within
).
All maximal ideals are
prime ideals,
[1] but not all prime ideals are maximal ideals.
[2] For example,
is a prime ideal in
, but it is not a maximal ideal because it is contained within all other ideals in
.
See also
Notes
- ↑ 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).
- ↑ John J. Watkins, Topics in Commutative Ring Theory. Princeton & Oxford: Princeton University Press (2007): p. 30, Corollary 3.1.
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