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Topic: Maximal ideal

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Prime ideal

Glossary of ring theory

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	Maximal ideal - Wikipedia, the free encyclopedia
		Maximal ideals are important because the quotient rings of maximal ideals are simple rings and in the special case of unital commutative rings even fields.
		In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number.
		Maximal ideals can be directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.
	en.wikipedia.org /wiki/Maximal_ideal (240 words)

	Prime ideal
		In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal.
		One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
		The introduction of prime ideals in algebraic number theory was a major step forward, since it made comprehensible the failure of the fundamental theorem of arithmetic.
	www.ebroadcast.com.au /lookup/encyclopedia/ma/Maximal_ideal.html (454 words)

	PlanetMath: existence of maximal ideals
		Note also that the use of the Axiom of Choice (in the form of Zorn's Lemma) is necessary, as there are models of ZF in which the above theorem and corollary fail.
		This is version 18 of existence of maximal ideals, born on 2003-09-08, modified 2006-11-13.
		After seeing this object (existence of maximal ideals) though, I realize that it could be adapted to supply a proof for the result that I wanted to add.
	planetmath.org /encyclopedia/EveryRingHasAMaximalIdeal.html (372 words)

	Springer Online Reference Works (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-20)
		This relation between the points of the interval and the maximal ideals has resulted in the construction of various theories for representing rings as rings of functions on a topological space.
		In a distributive lattice, as in a commutative ring, all maximal ideals are prime; the converse implication holds in a Boolean algebra, and indeed a distributive lattice in which all prime ideals are maximal is necessarily Boolean.
		The construction of maximal ideals in arbitrary rings or lattices generally requires an appeal to Zorn's lemma (see Axiom of choice or Zorn lemma), and indeed the maximal ideal theorem for many classes of rings or lattices (i.e.
	eom.springer.de.cob-web.org:8888 /m/m062960.htm (555 words)

	Ideal (ring theory) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-20)
		Ideals are important because they appear as the kernels of ring homomorphisms and allow one to define factor ring.
		Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J.
		The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given commutative ring forms a lattice.
	en.wikipedia.org.cob-web.org:8888 /wiki/Ideal_(ring_theory) (1224 words)

	Prime ideal (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-20)
		In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.
		In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e.
		Every maximal ideal is in fact prime; the converse is not true, in general.
	prime-ideal.iqnaut.net.cob-web.org:8888 (782 words)

	Ideals and Quotients
		For an ideal I the coefficient height is defined to be the maximum integer occurring in the current representation of the ideal: If the ideal is given via two elements, this will be the maximal coefficient height of the generators, otherwise the maximal entry of the basis matrix.
		For an ideal I the coefficient length is defined to be the size of the current representation: If the ideal is given via two elements, this will be the sum of the coefficient lengths of the generators, otherwise the sum of the entries of the basis matrix.
		All ideals of maximal orders can be generated by one or two elements of the field of fractions of the order they are an ideal of.
	www.math.lsu.edu /magma/text641.htm (3708 words)

	Springer Online Reference Works
		In particular, the maximal ideal space of a commutative Banach algebra is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.
		of continuous functions on the unit circle that have an analytic continuation inside the unit disc is a maximal subalgebra of the algebra of continuous functions on the unit circle.
		This algebra is a maximal subalgebra of the algebra of all continuous functions on the torus.
	eom.springer.de /c/c023380.htm (1448 words)

	Maximal ideals in the algebra of operators on Banach spaces (Site not responding. Last check: 2007-10-20)
		Maximal ideals in the algebra of operators on Banach spaces
		Maximal ideals in the algebra of operators on certain Banach spaces
		The Brown-McCoy radical of B(X), which by definition is the intersection of all maximal ideals in B(X), cannot be turned into an operator ideal.
	www.math.ku.dk /~laustsen/abstractmaxideals.html (234 words)

	PlanetMath: maximal ideal
		See Also: proper ideal, module, comaximal ideals, prime ideal, existence of maximal ideals
		Cross-references: field, quotient ring, commutative, prime ideals, simple ring, two-sided ideal, right ideal, simple, left ideal, proper subset, ideal, right, identity, ring
		This is version 3 of maximal ideal, born on 2001-10-20, modified 2002-04-20.
	planetmath.org /encyclopedia/MaximalIdeal.html (103 words)

	Prime ideal - Wikipedia, the free encyclopedia
		Every maximal ideal is in fact prime; the converse is true in a principal ideal domain, but is not true in general.
		The preimage of a prime ideal under a ring homomorphism is a prime ideal
		This is close to the historical point of view of ideals as ideal numbers, as "A is contained in P" is another way of saying "P divides A".
	en.wikipedia.org /wiki/Prime_ideal (804 words)

	No Title (Site not responding. Last check: 2007-10-20)
		0\}.$">Show that J is an ideal and is equal to the intersection of the prime ideals of A containing I.
		This maximal element is a maximal ideal and by uniqueness has to be M.
		We claim that M is the unique maximal ideal and hence A is local.
	www.math.gatech.edu /~saugata/teaching/fall00/sol5/sol5.html (806 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		R is a maximal ideal of R. Proposition.
		I is the intersection of all prime ideals of R that contain I. In any principal ideal domain, our next definitions both reduce to the statement that the ideal in question is generated by a power of an irreducible element.
		One important consequence of the generalized principal ideal theorem is that any Noetherian ring satisfies the descending chain condition for prime ideals.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Spectra of Rings
		This is a ring and contains maximal hence prime ideals by the zorn lemma, hence the pullback of any one such under the map R/A goes to the localization, is a prime ideal of R not containing A but not f.
		Then they proved that O(p) is a radical ideal, hence an interscetion of prime ideals, hence there exist other prime ideals that are not maximal.
		So the pullback to R of any maximal proper ideal of the localization of S at powers of f, is a prime ideal of R containing J but not f.
	www.physicsforums.com /showthread.php?p=284193 (1926 words)

	Springer Online Reference Works
		A function algebra is said to be analytic if all functions of this algebra that vanish on a non-empty open subset of the space of maximal ideals vanish identically.
		An ideal in a Banach algebra is said to be primary if it is contained in only one maximal ideal.
		-compact completely-regular space is homeomorphic to the Gleason part of the space of maximal ideals of some algebra, such that the restriction of the algebra to this part contains all bounded continuous functions.
	eom.springer.de /a/a011370.htm (968 words)

	The Radical of a Ring
		An ideal is called quasi-regular if all its elements are quasi-regular.
		An ideal is nil if all its elements are nilpotent.
		is intersections of kernels of ring homomorphisms hence a two-sided ideal.
	www.wisdom.weizmann.ac.il /~nadavs/reps/node6.html (145 words)

	CJM - Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$
		Let $m$ be a point of the maximal ideal space of $\papa$ with nontrivial Gleason part $P(m)$.
		We characterize the points $m$ for which $L_m$ is a homeomorphism in terms of interpolating sequences, and we show that in this case $\papa \circ L_m$ coincides with $\papa$.
		Also, if $I_m$ is the ideal of functions in $\papa$ that identically vanish on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.
	journals.cms.math.ca /cgi-bin/vault/view/suarez0942 (125 words)

	[No title]
		The corresponding maximal ideal in the polynomial ring
		be the maximal ideal in the local ring.
		Even though the tangent space is the same as that of the previous case, we want to think of these as different kinds of singularities; we need better invariants to do so.
	odin.mdacc.tmc.edu /~krc/agathos/local.html (851 words)

	[No title]
		X = Spec(R) is a noetherian topological space, since the closed subsets correspond to ideals.
		Since the original chain had maximal length, one of these inclusions must be an equality.
		In fact, however, both these ideals are primes of height 1, so they must be equal.
	odin.mdacc.tmc.edu /~krc/agathos/dimen.html (969 words)

	maximal - definition of maximal by the Free Online Dictionary, Thesaurus and Encyclopedia. (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-20)
		Of, relating to, or consisting of a maximum.
		An element in an ordered set that is followed by no other.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	www.thefreedictionary.com.cob-web.org:8888 /maximal (122 words)

	maximal - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "maximal" is defined.
		Phrases that include maximal: maximal ideal, maximal dose, maximal independent set, maximal torus, maximal histalog test, more...
		Words similar to maximal: maximally, maximum, extreme, more...
	www.onelook.com /?w=maximal (187 words)

	My maximal ideal Nissan Maxima QX :: AboutMyCar (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-20)
		My maximal ideal Nissan Maxima QX :: AboutMyCar (via CobWeb/3.1 planet03.csc.ncsu.edu)
		NIssan cars have no shortcomings that's why I decided to buy Nissan 1.5 year ago.
		You see my Nissan Maxima is a genuinly maximally ideal model with everything perfectly performed.
	www.aboutmycar.com.cob-web.org:8888 /stories/my-maximal-ideal-nissan-maxima-qx-941.htm (257 words)

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