Editing Maximal ideal

{{Short description|Ideal of a ring contained in no other ideal except the ring itself}}
In [[mathematics]], more specifically in [[ring theory]], a '''maximal ideal''' is an [[ideal (ring theory)|ideal]] that is [[maximal element|maximal]] (with respect to [[set inclusion]]) amongst all ''proper'' ideals.<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> In other words, ''I'' is a maximal ideal of a [[ring (mathematics)|ring]] ''R'' if there are no other ideals contained between ''I'' and ''R''.

Maximal ideals are important because the [[Quotient ring|quotients of rings]] by maximal ideals are [[simple ring]]s, and in the special case of [[Ring_(mathematics)#Notes_on_the_definition|unital]] [[commutative ring]]s they are also [[field (mathematics)|field]]s.  The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the [[Zariski topology]], is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. 

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]], 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 [[duality (mathematics)|dual notion]] to that of [[minimal ideal]]s.

==Examples==
* If '''F''' is a field, then the only maximal ideal is {0}.
* In the ring '''Z''' of integers, the maximal ideals are the [[principal ideal]]s generated by a prime number.
* More generally, all nonzero [[prime ideal]]s are maximal in a [[principal ideal domain]].
* The ideal <math> (2, x) </math> is a maximal ideal in ring <math> \mathbb{Z}[x] </math>. Generally, the maximal ideals of <math> \mathbb{Z}[x] </math> are of the form <math> (p, f(x)) </math> where <math> p </math> is a prime number and <math> f(x) </math> is a polynomial in <math> \mathbb{Z}[x] </math> which is irreducible modulo <math> p </math>.
* 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 <math> R </math> whenever there exists an integer <math> n > 1 </math> such that <math> x^n = x </math> for any <math> x \in R </math>.
* The maximal ideals of the [[polynomial ring]] <math>\mathbb{C}[x]</math> are principal ideals generated by <math>x-c</math> for some <math>c\in \mathbb{C}</math>.
* More generally, the maximal ideals of the polynomial ring {{nowrap|''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} over an [[algebraically closed field]] ''K'' are the ideals of the form {{nowrap|(''x''<sub>1</sub>&nbsp;&minus;&nbsp;''a''<sub>1</sub>, ..., ''x''<sub>''n''</sub>&nbsp;&minus;&nbsp;''a''<sub>''n''</sub>)}}. 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]]. This fact can fail in non-unital rings. For example, <math>4\mathbb{Z}</math> is a maximal ideal in <math>2\mathbb{Z} </math>, but <math>2\mathbb{Z}/4\mathbb{Z}</math> 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]] 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 ideal]]s for possible ways to circumvent this problem.
* In a commutative ring with unity, every maximal ideal is a [[prime ideal]]. The converse is not always true: for example, in any nonfield [[integral domain]] the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as [[Commutative ring#Dimension|zero-dimensional rings]], where the dimension used is the [[Krull dimension]].
* If ''k'' is a field, the preimage of a maximal ideal of a [[Finitely generated algebra|finitely generated ''k''-algebra]] under a ''k''-algebra homomorphism is a maximal ideal.  However, the preimage of a maximal ideal of a unital commutative ring under a ring homomorphism is not necessarily maximal.  For example, let <math>f:\mathbb{Z}\to\mathbb{Q}</math> be the inclusion map and <math>\mathfrak{n}=(0)</math> in <math>\mathbb{Q}</math>.  Then <math>\mathfrak{n}</math> is maximal in <math>\mathbb{Q}</math> but <math>f^{-1}(\mathfrak{n})=(0)</math> is not maximal in <math>\mathbb{Z}</math>.
* A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let <math>M_{n\times n}(\mathbb{Z})</math> be the ring of all <math>n\times n</math> matrices over <math>\mathbb{Z}</math>. This ring has a maximal ideal <math>M_{n\times n}(p\mathbb{Z})</math> for any prime <math>p</math>, but this is not a prime ideal since (in the case <math>n=2</math>)<math>A=\text{diag}(1,p)</math> and <math>B=\text{diag}(p,1)</math> are not in <math>M_{n\times n}(p\mathbb{Z})</math>, but <math>AB=pI_2\in M_{n\times n}(p\mathbb{Z})</math>. However, maximal ideals of noncommutative rings ''are'' prime in the [[#Generalization|generalized sense]] below.

==Generalization==
For an ''R''-module ''A'', a '''maximal submodule''' ''M'' of ''A'' is a submodule {{nowrap|''M'' ≠ ''A''}} satisfying the property that for any other submodule ''N'', {{nowrap|''M'' ⊆ ''N'' ⊆ ''A''}} implies {{nowrap|1=''N'' = ''M''}} or {{nowrap|1=''N'' = ''A''}}. Equivalently, ''M'' is a maximal submodule if and only if the quotient module ''A''/''M'' is a [[simple module]]. The maximal right ideals of a ring ''R'' are exactly the maximal submodules of the module ''R''<sub>''R''</sub>.

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]]s have maximal submodules.

As with rings, one can define the [[radical of a module]] 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 <sub>''R''</sub>''R''<sub>''R''</sub>.

==See also==
*[[Prime ideal]]

==References==
{{reflist}}
*{{citation  |author1=Anderson, Frank W.   |author2=Fuller, Kent R.  |title=Rings and categories of modules   |series=Graduate Texts in Mathematics   |volume=13   |edition=2   |publisher=Springer-Verlag  |place=New York   |year=1992   |pages=x+376   |isbn=0-387-97845-3  |mr=1245487 |doi=10.1007/978-1-4612-4418-9}}
*{{citation   |author=Lam, T. Y.   |title=A first course in noncommutative rings  |series=Graduate Texts in Mathematics   |volume=131   |edition=2   |publisher=Springer-Verlag   |place=New York   |year=2001   |pages=xx+385   |isbn=0-387-95183-0   |mr=1838439 |doi=10.1007/978-1-4419-8616-0}}

{{DEFAULTSORT:Maximal Ideal}}
[[Category:Ideals (ring theory)]]
[[Category:Ring theory]]
[[Category:Prime ideals]]