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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 In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
, 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 considere ...
that is maximal (with respect to
set inclusion In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
) 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 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 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 ...
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 In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
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 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 In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
over a field, the
zero ideal In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive ident ...
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 ideals.

# 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 it ...
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 ...
s are maximal in a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
. * The ideal $\left(2, x\right)$ is a maximal ideal in ring . Generally, the maximal ideals of are of the form $\left(p, f\left(x\right)\right)$ where $p$ is a prime number and $f\left(x\right)$ is a polynomial in which is irreducible modulo $p$. * 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 $R$ whenever there exists an integer $n > 1$ such that $x^n = x$ for any $x \in R$. * 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 variab ...
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
''K'' are the ideals of the form . This result is known as the weak Nullstellensatz.

# Properties

* An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals. * If ''R'' is a unital commutative ring with an ideal ''m'', then ''k'' = ''R''/''m'' is a field if and only if ''m'' is a maximal ideal. In that case, ''R''/''m'' is known as the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is ...
. This fact can fail in non-unital rings. For example, $4\mathbb$ is a maximal ideal in $2\mathbb$, but $2\mathbb/4\mathbb$ is not a field. * If ''L'' is a maximal left ideal, then ''R''/''L'' is a simple left ''R''-module. Conversely in rings with unity, any simple left ''R''-module arises this way. Incidentally this shows that a collection of representatives of simple left ''R''-modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of ''R''. * Krull's theorem (1929): Every nonzero unital ring has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero
finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts in ...
has a maximal submodule. Suppose ''I'' is an ideal which is not ''R'' (respectively, ''A'' is a right ideal which is not ''R''). Then ''R''/''I'' is a ring with unity (respectively, ''R''/''A'' is a finitely generated module), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal (respectively, maximal right ideal) of ''R'' containing ''I'' (respectively, ''A''). * Krull's theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals. See regular ideals for possible ways to circumvent this problem. * In a commutative ring with unity, every maximal ideal is a
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 ...
. The converse is not always true: for example, in any nonfield
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension. * A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let $M_\left(\mathbb\right)$ be the ring of all $n\times n$ matrices over $\mathbb$. This ring has a maximal ideal $M_\left(p\mathbb\right)$ for any prime $p$, but this is not a prime ideal since (in the case $n=2$)$A=\text\left(1,p\right)$ and $B=\text\left(p,1\right)$ are not in $M_\left(p\mathbb\right)$, but $AB=pI_2\in M_\left(p\mathbb\right)$. However, maximal ideals of noncommutative rings ''are'' prime in the generalized sense below.

# Generalization

For an ''R''-module ''A'', a maximal submodule ''M'' of ''A'' is a submodule satisfying the property that for any other submodule ''N'', implies or . Equivalently, ''M'' is a maximal submodule if and only if the quotient module ''A''/''M'' is a
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
. The maximal right ideals of a ring ''R'' are exactly the maximal submodules of the module ''R''''R''. Unlike rings with unity, a nonzero module does not necessarily have maximal submodules. However, as noted above, ''finitely generated'' nonzero modules have maximal submodules, and also
projective module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characteriz ...
s have maximal submodules. As with rings, one can define the
radical of a module In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle s ...
using maximal submodules. Furthermore, maximal ideals can be generalized by defining a maximal sub-bimodule ''M'' of a bimodule ''B'' to be a proper sub-bimodule of ''M'' which is contained in no other proper sub-bimodule of ''M''. The maximal ideals of ''R'' are then exactly the maximal sub-bimodules of the bimodule ''R''''R''''R''.