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Topic: Prime ideal

Related Topics

Ring ideal

Principal ideal

Dedekind domain

Principal ideal domain

Local ring

Commutative ring

Ring (mathematics)

Prime number

Prime

Integral domain

Euclidean domain

Composite number

Finitely generated group

Integer

Algebra over a field

	Prime ideal - Wikipedia, the free encyclopedia
		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.
		One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
		An ideal such that ab in P implies that a or b is in P is called a completely prime ideal.
	en.wikipedia.org /wiki/Prime_ideal (785 words)

	Boolean prime ideal theorem - Wikipedia, the free encyclopedia
		Although the various prime ideal theorems may appear simple and intuitive, they can in general not be derived from the axioms of Zermelo-Fraenkel set theory (ZF).
		Historically, the first statement relating to later prime ideal theorems was in fact referring to filters -- subsets that are ideals with respect to the dual order.
		The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras.
	en.wikipedia.org /wiki/Boolean_prime_ideal_theorem (1671 words)

	Encyclopedia: Prime ideal (Site not responding. Last check: 2007-11-07)
		If R is a commutative ring, then an ideal P of R is prime if it has the following two properties: In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law.
		In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
		If R is a noncommutative ring, then an ideal P of R is prime if it has the following two properties: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers.
	www.nationmaster.com /encyclopedia/prime-ideal (1919 words)

	Prime ideal: Definition and Links by Encyclopedian.com - All about Prime ideal (Site not responding. Last check: 2007-11-07)
		A natural number n is a prime number if and only if the ideal Zn is a prime ideal in Z.
		A commutative ring is a field if and only if {0} is its only prime ideal, or alternatively, if and only if {0} is a maximal ideal.
		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.encyclopedian.com /pr/Prime-ideal.html (485 words)

	Prime ideal -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Prime ideals in (Click link for more info and facts about order theory) order theory are treated in the article on (Click link for more info and facts about ideals in order theory) ideals in order theory.
		Every maximal ideal (see above) is prime; an ideal I in the commutative ring R is a maximal ideal if and only if the factor ring R/I is a (A piece of land cleared of trees and usually enclosed) field.
		One use of prime ideals occurs in (Click link for more info and facts about algebraic geometry) algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
	www.absoluteastronomy.com /encyclopedia/p/pr/prime_ideal.htm (881 words)

	Prime ideal - Definition up Erdmond.Com
		A prime ideal is an ideal of a commutative or noncommutative ring.
		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 does not work in rings of algebraic_integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind_domain.
	www.erdmond.com /Prime_ideal.html (789 words)

	PlanetMath: examples of prime ideal decomposition in number fields (Site not responding. Last check: 2007-11-07)
		"examples of prime ideal decomposition in number fields" is owned by alozano.
		, prime ideal decomposition in cyclotomic extensions of
		This is version 9 of examples of prime ideal decomposition in number fields, born on 2003-08-20, modified 2003-08-27.
	planetmath.org /encyclopedia/ExamplesOfPrimeIdealDecompositionInNumberFields.html (239 words)

	[No title]
		Recall that the height of an ideal I is defined as ht (I) = min{ht (p)I H p, p F H * prime }.
		We denote by Proj (H ) the spectrum of homogeneous prime* ideals in H , and by Proj P (H ) the spectrum of P invariant homogeneous prime ideals, where an ideal is called P * invariant if it is stable under the action of the Steenrod algebra P *.
		Since this is possible for every prime ideal q of height k, this means that no prime ideal of height k is contained in a prime ideal associated to (d q s n,0,.
	hopf.math.purdue.edu /Neusel/uncoma.txt (4896 words)

	Prime ideal (Site not responding. Last check: 2007-11-07)
		This article only covers ideals of ring theory.
		It turns out that the irreducible varieties correspond to prime ideals.
		The zero ideal of any prime ring is prime.
	www.sciencedaily.com /encyclopedia/prime_ideal (823 words)

	PlanetMath: minimal prime ideal (Site not responding. Last check: 2007-11-07)
		is called a minimal prime ideal if it does not properly contain any other prime ideal of
		is a prime ring, then the zero ideal is a prime ideal, and is thus the unique minimal prime ideal of
		This is version 2 of minimal prime ideal, born on 2001-11-23, modified 2003-09-20.
	planetmath.org /encyclopedia/MinimalPrimeIdeal.html (84 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In abstract algebra, the concept of prime ideals is an important generalization of the concept of prime numbers.
		A positive integer n is a prime number if and only if the ideal Zn is a prime ideal in Z.
		The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the ordinary fundamental theorem of arithmetic does not work in rings of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.
	www.informationgenius.com /encyclopedia/p/pr/prime_ideal.html (496 words)

	► » Prime ideals in comm. ring - HELP! (Site not responding. Last check: 2007-11-07)
		Yes, the intersection and union of a chain of ideals is an ideal.
		Zorn's lemma: the intersection of a chain of prime ideals is prime,
		Also the union of a chain of prime ideals is a prime ideal.
	www.science-chat.org /Prime-ideals-in-comm-ring--HELP-6713883.html (1073 words)

	Ideals and Quotients
		Given an integral ideal I of O, return two elements of the field of fractions of O that form a two-element normal presentation for I, as well as an integer g such that I is g-normal.
		The denominator of the fractional ideal I. This is the smallest positive integer d such that dI is an integral ideal.
		Returns the basis matrix for the ideal I of O. The basis matrix consists of the elements of a Z-basis for the ideal written as rows of rational coefficients with respect to the power basis of the number field K of which O is an order.
	www.math.uiuc.edu /Software/magma/text359.html (882 words)

	d(r)
		A prime ideal p in r is in the set d(r) if it is the smallest prime ideal containing the annihilator [0:x] for some x.
		If there is a prime ideal p containing [0:x] for some x, p descends to a minimal prime ideal containing [0:x], a member of d(r).
		Each associate is prime, and when a radical is prime, it is the minimal prime ideal containing [0:x].
	www.mathreference.com /id-pry,dr.html (809 words)

	prime_ideal (Site not responding. Last check: 2007-11-07)
		Since each prime ideal obviously is an ideal, you can do everything you can do with an alg_ideal also with a prime_ideal.
		A prime_ideal consists of a prime p (represented by a bigint) and an alg_number a which especially contains the pointer to an nf_base B, such that B is the basis of the order O over which the prime ideal is defined and that p and a generate the prime ideal as O-ideal.
		Additionaly the lidia_size_ts e and f contain the ramification index and the degree of inertia of the prime ideal, respectively.
	www.math.psu.edu /local_doc/LiDIA/node97.html (464 words)

	Integrally Closed and One Prime Ideal
		Because r is noetherian, it is also a pid, and a dvr with valuation group Z. The ideals are precisely the powers of the maximal ideal, and the maximal ideal is the only prime ideal.
		Start with the ideal generated by x, and find a larger ideal in s, then a larger one, then a larger one, and so on, until the process stops, which it must, since r is noetherian.
		Every ideal is a power of m, ideals are linearly ordered, and r is a valuation ring.
	www.mathreference.com /id-dd,intcl.html (878 words)

	Boolean prime ideal theorem (Site not responding. Last check: 2007-11-07)
		In mathematics, a number of so called prime ideal theorems for guaranteeing the existence of certain subsets of an abstract algebra can be stated.
		The prototypical properties that were disscussed for Boolean algebras in the above section can easily be modified to include more general lattices, such as distributive lattices or Heyting algebras.
		M. Erné, Prime Ideal Theory for General Algebras, Applied Categorical Structures 8, 115--144, 2000.
	www.sciencedaily.com /encyclopedia/boolean_prime_ideal_theorem (1707 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings (Site not responding. Last check: 2007-11-07)
		An integral domain D is called a Dedekind domain if each proper ideal of D can be written as a product of a finite number of prime ideals of D. We will show in Theorem 12.2.4 that a Dedekind domain has some of the properties of a principal ideal domain.
		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)

	ABSTRACT ALGEBRA ON LINE: Structure of Noncommutative Rings (Site not responding. Last check: 2007-11-07)
		P, for any ideals A, B of R. A proper ideal I of the ring R is called a semiprime ideal if it is an intersection of prime ideals of R. A proper ideal P of the ring R is called a left primitive ideal if it is the annihilator of a simple left R-module.
		In a left Artinian ring, the notions of maximal ideal, primitive ideal, and prime ideal coincide.
		If P is a primitive ideal of the ring R, then there exists a division ring D and a vector space V over D for which R/P is isomorphic to a subring of the ring of all linear transformations from V into V. Proposition.
	www.math.niu.edu /~beachy/aaol/noncommutative.html (1031 words)

	Prime Ideals are Maximal
		If u is an ideal inside a prime ideal m, there is a minimal prime ideal p containing u and contained in m.
		Because of the smaller prime ideal, m is not principle.
		Since maximal ideals are prime, we have a prime containing a prime, which is impossible.
	www.mathreference.com /id-dd,primax.html (1130 words)

	Finite Rings (Site not responding. Last check: 2007-11-07)
		We can also see this by noting that the ideal consisting of all nilpotent elements of a ring is the intersection of all its prime ideals.
		We have thus shown that any non-nilpotent element is in the complement of some prime ideal; the converse, that a nilpotent element belongs to all prime ideals, is also clear.
		Hence the intersection of all prime ideals is also the ideal consisting of all nilpotent elements of a ring; this ideal is called the nil radical of the ring.
	www.imsc.ernet.in /~kapil/geometry/caag/finite.html (2519 words)

	Prime article - Prime Prime element Prime ideal Prime lens Prime number Prime rate - What-Means.com (Site not responding. Last check: 2007-11-07)
		Prime article - Prime Prime element Prime ideal Prime lens Prime number Prime rate - What-Means.com
		This is a disambiguation page; that is, one that points to other pages that might otherwise have the same name.
		Prime article - Prime definition - what means Prime
	www.what-means.com /encyclopedia/Prime (81 words)

	Boolean prime ideal theorem (Site not responding. Last check: 2007-11-07)
		An ideal in a Boolean algebra A is a subset I of A such that
		It is an exercise to show that this concept of ideal is then no different from the concept of ring ideal known to every student of abstract algebra.
		In Boolean algebras, unlike rings in general, there is no difference between a prime ideal and a maximal ideal.
	www.ukpedia.com /b/boolean-prime-ideal-theorem.html (234 words)

	PlanetMath: prime ideal (Site not responding. Last check: 2007-11-07)
		is called a prime ideal if the following equivalent conditions are met:
		Cross-references: integral domain, quotient ring, prime, identity, commutative, prime ring, ideals, right ideals, satisfies, product of ideals, left ideals, equivalent, proper ideal, ring
		This is version 10 of prime ideal, born on 2001-10-20, modified 2005-07-24.
	planetmath.org /encyclopedia/PrimeIdeal.html (111 words)

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