In
mathematics, more specifically in
ring theory, a maximal ideal is an
ideal that is
maximal (with respect to
set inclusion) amongst all ''proper'' ideals.
In other words, ''I'' is a maximal ideal of a
ring ''R'' if there are no other ideals contained between ''I'' and ''R''.
Maximal ideals are important because the
quotients of rings by maximal ideals are
simple rings, and in the special case of
unital commutative rings they are also
fields.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a
simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a
local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definitio ...
J(''R'').
It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2
square matrices over a field, the
zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.
Definition
There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring ''R'' and a proper ideal ''I'' of ''R'' (that is ''I'' ≠ ''R''), ''I'' is a maximal ideal of ''R'' if any of the following equivalent conditions hold:
* There exists no other proper ideal ''J'' of ''R'' so that ''I'' ⊊ ''J''.
* For any ideal ''J'' with ''I'' ⊆ ''J'', either ''J'' = ''I'' or ''J'' = ''R''.
* The quotient ring ''R''/''I'' is a simple ring.
There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal ''A'' of a ring ''R'', the following conditions are equivalent to ''A'' being a maximal right ideal of ''R'':
* There exists no other proper right ideal ''B'' of ''R'' so that ''A'' ⊊ ''B''.
* For any right ideal ''B'' with ''A'' ⊆ ''B'', either ''B'' = ''A'' or ''B'' = ''R''.
* The quotient module ''R''/''A'' is a simple right ''R''-module.
Maximal right/left/two-sided ideals are the
dual notion to that of
minimal ideal In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring ''R'' is a nonzero right ideal which contains no other nonzero right ideal. Likewise, a minimal left ideal is a nonzero left ideal of ''R'' containing no oth ...
s.
Examples
* If F is a field, then the only maximal ideal is .
* In the ring Z of integers, the maximal ideals are the
principal ideals generated by a prime number.
* More generally, all nonzero
prime ideals are maximal in a
principal ideal domain.
* The ideal
is a maximal ideal in ring
. Generally, the maximal ideals of
are of the form
where
is a prime number and
is a polynomial in
which is irreducible modulo
.
* Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring
whenever there exists an integer
such that
for any
.
* The maximal ideals of the
polynomial ring