71.41.210.146: /* See also */ Varicode

2014-11-10T06:48:56Z

216.66.5.43: /* Generalizations */ Link to Tribonacci numbers

2014-02-06T15:21:38Z

Generalizations: Link to Tribonacci numbers

124.170.241.132: /* Example */ Why python?

2014-01-24T10:51:59Z

Example: Why python?

← Older revision		Revision as of 12:51, 24 January 2014
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	I'~~m Aracelis~~ and ~~I live~~ in a ~~seaside city~~ in ~~northern Italy~~, ~~Ballino~~. I'~~m 24 and~~ I'm will ~~soon finish my study at~~ [~~https:~~//~~www~~.~~google~~.~~com~~/~~search?hl=en~~&gl=~~us&tbm~~=~~nws&q~~=~~Agriculture&btnI~~=~~lucky Agriculture~~] and [~~http~~://~~Browse~~.~~deviantart~~.~~com/~~?qh=~~&section~~=~~&global~~=~~1&q~~=~~Life+Sciences Life Sciences~~].<br><br>~~Feel free~~ to ~~visit my web blog; [http:~~//~~www~~.~~200box~~.com/%E0%B8%95%E0%B8%B9%E0%B9%89%E0%B8%84%E0%B8%AD%E0%B8%99%E0%B9%80%E0%B8%97%E0%B8%99%E0%B9%80%E0%B8%99%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%AA%E0%B8%B3%E0%B9%80%E0%B8%A3%E0%B9%87%E0%B8%88%E0%B8%A3%E0%B8%B9/ บ้านตู้คอนเทนเนอร์]		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''.

			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.

			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 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.

			==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==
			* 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 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]].

			==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 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'').

			* 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]].
			<!--
			* (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.-->

			==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>.

			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.

			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 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>.

			==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 }}
			*{{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 }}

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

en>Gadfium: rv, unexplained deletion of content.

2012-07-23T04:11:43Z

rv, unexplained deletion of content.

New page

I'm Aracelis and I live in a seaside city in northern Italy, Ballino. I'm 24 and I'm will soon finish my study at [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=Agriculture&btnI=lucky Agriculture] and [http://Browse.deviantart.com/?qh=&section=&global=1&q=Life+Sciences Life Sciences].<br><br>Feel free to visit my web blog; [http://www.200box.com/%E0%B8%95%E0%B8%B9%E0%B9%89%E0%B8%84%E0%B8%AD%E0%B8%99%E0%B9%80%E0%B8%97%E0%B8%99%E0%B9%80%E0%B8%99%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%AA%E0%B8%B3%E0%B9%80%E0%B8%A3%E0%B9%87%E0%B8%88%E0%B8%A3%E0%B8%B9/ บ้านตู้คอนเทนเนอร์]

Fibonacci coding - Revision history

71.41.210.146: /* See also */ Varicode

216.66.5.43: /* Generalizations */ Link to Tribonacci numbers

124.170.241.132: /* Example */ Why python?

en>Gadfium: rv, unexplained deletion of content.