Maximal ideal
In ring theory, a maximal ideal of a ring is a proper ideal
which is not contained in any other proper ideal of
. (That is,
, and there is no ideal
with
.)
One important property of maximal ideals is that if is a commutative ring with unity then the quotient ring
is a field iff
is a maximal ideal of
.
From this it follows that in a commutative ring with unity that any maximal ideal is prime. Indeed if is a maximal ideal of a commutative ring with unity
, then by the above observation
is a field. But then
must clearly be an integral domain, and this happens iff
is prime. Hence
is indeed a prime ideal of
.
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