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| In [[mathematics]], more specifically in [[ring theory]], a '''maximal ideal''' is an [[ideal (ring theory)|ideal]] which 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 ''R'' if there are no other ideals contained between ''I'' and ''R''.
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| Maximal ideals are important because the [[quotient ring]]s of maximal ideals are [[simple ring]]s, and in the special case of [[unital algebra|unital]] [[commutative ring]]s they are also [[field (mathematics)|field]]s.
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| 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'').
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| It is possible for a ring to have a unique maximal 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 ideal, but there are many maximal right ideals.
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| ==Definition==
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| 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:
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| * There exists no other proper ideal ''J'' of ''R'' so that ''I'' ⊊ ''J''.
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| * For any ideal ''J'' with ''I'' ⊆ ''J'', either ''J'' = ''I'' or ''J'' = ''R''.
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| * The quotient ring ''R''/''I'' is a simple ring.
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| 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'':
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| * There exists no other proper right ideal ''B'' of ''R'' so that ''A'' ⊊ ''B''.
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| * For any right ideal ''B'' with ''A'' ⊆ ''B'', either ''B'' = ''A'' or ''B'' = ''R''.
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| * The quotient module ''R''/''A'' is a simple right ''R'' module.
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| Maximal right/left/two-sided ideals are the [[duality (mathematics)|dual notion]] to that of [[minimal ideal]]s.
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| ==Examples==
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| * In the ring '''Z''' of integers the maximal ideals are the [[principal ideal]]s generated by a prime number.
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| * More generally, all nonzero [[prime ideal]]s are maximal in a [[principal ideal domain]].
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| * 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> − ''a''<sub>1</sub>,...,''x''<sub>''n''</sub> − ''a''<sub>''n''</sub>)}}. This result is known as the weak [[nullstellensatz]].
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| ==Properties==
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| * An important ideal of the ring called the [[Jacobson radical]] can be defined using maximal right (or maximal left) ideals.
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| * 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.
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| * 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''.
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| * '''[[Krull's theorem]]''' (1929): Every ring with a multiplicative identity 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'').
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| * 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.
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| * 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]].
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| <!--
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| * (This sentence is not clear: Lattice diagram of what poset? Does it need to be a commutative ring? "biggest containing ring"? "Prime property?") In a [[Hasse diagram|lattice diagram]], maximal ideals are always directly joined to the biggest containing ring, as follows from the prime property.-->
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| ==Generalization== | |
| For an ''R'' module ''A'', a '''maximal submodule''' ''M'' of ''A'' is a submodule ''M''≠''A'' for which for any other submodule ''N'', if ''M''⊆''N''⊆''A'' then ''N''=''M'' or ''N''=''A''. Equivalently, ''M'' is a maximal submodule if and only if the quotient module ''A''/''M'' is a [[simple module]]. Clearly the maximal right ideals of a ring ''R'' are exactly the maximal submodules of the module ''R''<sub>''R''</sub>.
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| Unlike rings with unity however, ''a 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.
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| As with rings, one can define the [[radical of a module]] using maximal submodules.
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| 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 by no other proper sub-bimodule of ''M''. So, the maximal ideals of ''R'' are exactly the maximal sub-bimodules of the bimodule <sub>''R''</sub>''R''<sub>''R''</sub>.
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| ==References==
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| {{reflist}}
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| *{{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 }}
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| *{{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 }}
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| {{DEFAULTSORT:Maximal Ideal}}
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| [[Category:Ideals]]
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| [[Category:Ring theory]]
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| [[Category:Prime ideals]]
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