In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
ring theory, a maximal ideal is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
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 ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
s, and in the special case of
unital commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s they are also
fields.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the
poset 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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebrai ...
, 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 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 other ...
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 ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
s generated by a prime number.
* More generally, all nonzero
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s 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
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...