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Topic: Principal ideal domain

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Principal ideal

Ring ideal

Prime ideal

Integral domain

Noetherian ring

Euclidean domain

Ring (mathematics)

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Commutative ring

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	Principal ideal domain (Site not responding. Last check: 2007-11-05)
		In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element).
		Every principal ideal domain is a unique factorization domain (UFD).The converse does not hold since for any field K, K[X,Y] is a UFD but is not a PID (to prove this look at the ideal generated by.
		An example of a principal ideal domain that is not a euclidean domain is the ring (Wilson, J. "A Principal Ring that is Not a Euclidean Ring." Math.
	www.encyclopedia-1.com /p/pr/principal_ideal_domain.html (287 words)

	Knowledge King - Ring ideal (Site not responding. Last check: 2007-11-05)
		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 ring forms a lattice.
		The term "ideal" comes from "ideal number": ideals were seen as a generalization of the concept of number.
		In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.
	www.knowledgeking.net /encyclopedia/r/ri/ring_ideal.html (1308 words)

	Principal ideal domain - Wikipedia, the free encyclopedia
		It is not principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.
		The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain.
		An example of a principal ideal domain that is not a Euclidean domain is the ring
	www.wikipedia.org /wiki/Principal_ideal_domain (306 words)

	Encyclopedia: Principal ideal domain
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers.
		In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...
		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.
	www.nationmaster.com /encyclopedia/Principal-ideal-domain (747 words)

	Principal ideal domain -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		In a principal ideal domain, any two elements have a (The largest integer that divides without remainder into a set of integers) greatest common divisor, and almost always have more than one.
		Every principal ideal domain is a (Click link for more info and facts about unique factorization domain) unique factorization domain (UFD).The converse does not hold since for any field K, K[X,Y] is a UFD but is not a PID (to prove this look at the ideal generated by.
		An example of a (Click link for more info and facts about principal ideal domain) principal ideal domain that is not a (Click link for more info and facts about euclidean domain) euclidean domain is the ring (Wilson, J. "A Principal Ring that is Not a Euclidean Ring." Math.
	www.absoluteastronomy.com /encyclopedia/p/pr/principal_ideal_domain.htm (342 words)

	PlanetMath: PID
		is an integral domain where every ideal is a principal ideal.
		See Also: UFD, irreducible, ideal, integral domain, Euclidean domain, Euclidean valuation, proof that an Euclidean domain is a PID, motivation for Euclidean domains
		This is version 3 of PID, born on 2001-11-04, modified 2005-01-30.
	planetmath.org /encyclopedia/PID.html (119 words)

	Principal ideal (Site not responding. Last check: 2007-11-05)
		In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a singleelement a of R.
		The ideal (x,y)generated by x and y, which consists of all the polynomials in C[x,y] that have zero for the constant term, is not principal.
		For a Dedekind domain R, we may also ask, given anon-principal ideal I of R, whether there is some extension S of R such that the ideal of Sgenerated by I is principal (said more loosely, I becomes principal in S).
	www.therfcc.org /RFCC/principal-ideal-210826.html (489 words)

	Encyclopedia: Principal ideal
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
		In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients.
		In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain.
	www.nationmaster.com /encyclopedia/Principal-ideal (1459 words)

	Ideal class group - Info Voyager : Travel Guides : Information Portal (Site not responding. Last check: 2007-11-05)
		The first ideal class groups encountered in mathematics were part of the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms.
		In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains).
		Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the image of I is a principal ideal in L.
	www.infovoyager.com /info/id/Ideal_class_group.html (1406 words)

	Euclidean domain
		More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements of D to non-negative integers and possessing the following properties:
		Every Euclidean domain is a principal ideal domain.
		In fact, if I is a nonzero ideal of a Euclidean domain D and a nonzero a in I is chosen to minimize g(a), then I = aD.
	www.ebroadcast.com.au /lookup/encyclopedia/eu/Euclidean_ring.html (263 words)

	Unique factorization domain -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		Any two (or finitely many) elements of a UFD have a (The largest integer that divides without remainder into a set of integers) greatest common divisor and a (The smallest multiple that is exactly divisible by every member of a set of numbers) least common multiple.
		A (Click link for more info and facts about Noetherian) Noetherian integral domain is a UFD if and only if every (The vertical dimension of extension; distance from the base of something to the top) height 1 (Click link for more info and facts about prime ideal) prime ideal is principal.
		An integral domain is a UFD if and only if the ascending chain condition holds for principal ideals, and any two elements of A have a least common multiple.
	www.absoluteastronomy.com /encyclopedia/U/Un/Unique_factorization_domain.htm (789 words)

	KR: PID (Site not responding. Last check: 2007-11-05)
		If I is an ideal in R, b is an element in R, and I is the set of b times every element in R, then I is a principal ideal.
		A domain is a commutative ring R where the cancellation law holds for multiplication.
		A domain R is called a principal ideal domain (PID) if every ideal I in R is a principal ideal.
	www.csl.mtu.edu /~kjrokos/PID.html (340 words)

	Principal ideal domain
		In a principal ideal domain, any two elements have a greatest common divisor (and may have more than one).
		Every principal ideal domain is Noetherian and a unique factorization domain.
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Principal_ideal_domain.html (138 words)

	Integer - Wikipédia
		This observation is the base for the Euclidean algorithm for computing greatest common divisors.
		All of this can be abbreviated by saying that Z is a Euclidean domain.
		This implies that Z is a principal ideal domain and that whole numbers can be written as products of primes in an essentially unique way.
	su.wikipedia.org /wiki/Integer (546 words)

	PlanetMath: finitely generated modules over a principal ideal domain
		be a finitely generated module over a principal ideal domain
		"finitely generated modules over a principal ideal domain" is owned by Thomas Heye.
		This is version 7 of finitely generated modules over a principal ideal domain, born on 2003-09-01, modified 2003-09-03.
	planetmath.org /encyclopedia/FinitelyGeneratedModulesOverAPrincipalIdealDomain.html (241 words)

	ABSTRACT ALGEBRA ON LINE: Unique Factorization
		Let D be a principal ideal domain, and let p be a nonzero element of D. Then p is irreducible in D if and only if pD is a prime ideal of D. Definition.
		Any principal ideal domain is a unique factorization domain.
		Let D be a unique factorization domain, let Q be the quotient field of D, and let f(x) be a primitive polynomial in D[x].
	www.math.niu.edu /~beachy/aaol/unique.html (552 words)

	[No title]
		Over a Sylvester domain every full matrix is regular, so the conditions of Theorem 5, and their duals, hold, so every flat module is spacial, hence locally free, and the kernel of any homomorphism between spacial modules is spacial, which says that the kernel of any homomorphism between flat modules is flat.
		Let R be a principal ideal domain and S be the set of atoms of R. Then inverting the elements of S gives the field of fractions, K, of R. By Theorem 13, R(X)-K(X) is rank-preserving, but K(X) is a sernifir so has a rank-preserving homomorphism to a skew field, and hence so does R~X).
		The conditions (l)-(4) for P to be a prime matrix ideal are readily verified using (35), (36), and the "transpose" of (36).
	www.math.rutgers.edu /~sontag/FTP_DIR/wdicks.txt (11071 words)

	[No title]
		Based on the characterisaion in Theorem 3, we give an algorithm for constructing strong Gröbner bases over a principal ideal ring, generalising thus Buchberger's results to a this type of ring.
		We use the fact that a ring is a principal ideal ring if and only if it is isomorphic to a direct product of principal ideal domains and Artinian chain rings.
		Using Theorem 4 these results can be generalised to principal ideal rings and the restriction on the length of the code can be removed.
	www-calfor.lip6.fr /ICPSS/papers/40Sa/40Sa.htm (959 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		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)

	Principal ideal - Wikipedia, the free encyclopedia
		a left principal ideal of R is a subset of R of the form Ra := {ra : r in R};
		a two-sided principal ideal is a subset of the form RaR := {r
		More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication.
	en.wikipedia.org /wiki/Principal_ideal (522 words)

	Graduate Study in Algebra
		One shows such rings are principal ideal domains and proves there is unique factorization of polynomials into products of irreducible polynomials.
		The goal of the course is the fundamental theorem of Galois theory and the solutions to the three pearls of antiquity: the quadrature of the circle, the trisection of an angle, and the duplication of the cube.
		The following topics are studied: the isomorphism theorems for groups, solvability of p-groups, simplicity of the alternating group on at least 5 letters, Sylow theorems, Jordan-Holder Theorem, principal ideal domains, Gauss' lemma, Eisenstein's criterion, the fundamental theorem of Galois theory, finite fields, cyclotomic fields, solvability of equations by radicals.
	www.math.uiuc.edu /GraduateProgram/researchmath/gradalgebra.html (1660 words)

	Dominio ideal principal (Site not responding. Last check: 2007-11-05)
		En álgebra abstracta, un dominio ideal principal (PID) es un dominio integral en el cual cada ideal es principal (es decir, generado por un solo elemento).
		En un dominio ideal principal, cualquier dos elementos tienen un divisor común más grande, y tienen casi siempre más de uno.
		En dominios ideales principales un inverso cercano sujeta: cada ideal primero distinto a cero es máximo.
	www.yotor.net /wiki/es/do/Dominio%20ideal%20principal.htm (342 words)

	Principal ideal domain (Site not responding. Last check: 2007-11-05)
		The ring Z[X] of all polynomials with integer coefficients isn't principal, since for example the ideal generated by 2 and X cannot be generated by a single polynomial.
		All is still licensed under the GNU FDL.
		He composed himself to rest and eat some so that he could safely give the horses a drink.
	www.termsdefined.net /pr/principal-ideal-domain.html (368 words)

	Corrections
		Corollary: A factorial domain R of dimension 1 is a principal ideal domain.
		On page 69: The analogue of Proposition 8.2, where principal ideal is replaced by finitely generated ideal, is true (see for instance [Mat2], Theorem 3.4).
		On page 91, Proposition III.2.7 states that a local noetherian integrally closed domain R of dimension 1 is a principal ideal domain.
	www.math.uga.edu /~lorenz/corrections.html (1202 words)

	► » Re: Zorn's Lemma (Site not responding. Last check: 2007-11-05)
		ideal of a principal ideal domain is contained in some maximal ideal.
		increasing chain of ideals in D is finite.
		the ideal I. Is it a maximal ideal in D? Probably not.
	www.science-chat.org /Re-Zorns-Lemma-8722090.html (452 words)

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