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Topic: Integral closure

	Integral closure - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-08)
		In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers.
		An equivalent definition is that R is integrally closed in S iff the integral closure of R in S is equal to R (in general the integral closure is a superset of R).
		The integral closure of Z in the complex numbers C is the set of all algebraic integers.
	en.wikipedia.org /wiki/Integral_closure (496 words)

	Closure (mathematics) - Wikipedia, the free encyclopedia
		For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers.
		The closure is idempotent: the closure of the closure equals the closure.
		In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X is closed under the operation of linear combination.
	en.wikipedia.org /wiki/Closure_(mathematics) (982 words)

	Integral closure: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-08)
		The terminology is justified by the fact that the integral closure of R in S is always integrally closed in S, EHandler: no quick summary.
		The integral closure of Z in the complex number[For more info, click on this link]s C is the set of all algebraic integer algebraic integer quick summary:
		In mathematics, an algebraic closure of a field k is an algebraic extension of k that is algebraically closed....
	www.absoluteastronomy.com /encyclopedia/i/in/integral_closure.htm (1122 words)

	PlanetMath: integral domain
		It is also not very closely related to the notion of integral, which is applied to ring elements, or that of integral closure, which is applied to extensions of rings, although these concepts are normally applied to integral domains.
		Commutative integral domains have fraction fields, which play the role of the rational numbers, and they each have a characteristic (which is either a prime number or zero).
		This is version 10 of integral domain, born on 2001-10-19, modified 2004-04-24.
	planetmath.org /encyclopedia/IntegralDomain.html (202 words)

	Algebraic and integral closures
		The algebraic closure of F in K is the subset of K consisting of elements algebraic over F. The subfield F of K is said to be algebraically closed in K if it is its own algebraic closure in K. Theorem.
		The integral closure of A in B is the subset of B consisting of elements integral over A; the subring A of B is said to be integrally closed in B if it is its own integral closure in B. Theorem (4.23).
		Let A be a subring of B. The integral closure of A in B is a ring, and is integrally closed in B. The proof is much the same as for algebraic closures.
	www.math.harvard.edu /~elkies/M250.04/closure.html (614 words)

	Thermoplastic bag closure - Patent 5669504
		Integral closure tabs are formed in the bag front and rear walls at the bag mouth.
		The rear wall closure tab is then pulled in the opposite direction across the bag mouth towards the front of the bag to engage an adhesive zone positioned on the exterior surface of the front wall.
		The front closure tab 50 and the rear closure tab 52 are separated by the distance between the front wall and the rear wall.
	www.freepatentsonline.com /5669504.html (5032 words)

	Closed in the Polynomial Ring
		If c is the integral closure of r in s, then the same is true of the corresponding polynomial rings; c[x] is the integral closure of r[x] in s[x].
		This means the coefficients of f (or g) lie in the integral closure of c in e.
		In summary, the integral closure of r[x] in s[x] is c[x].
	www.mathreference.com /id-ext,poly.html (718 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		As used herein, the closure zone 26 is a region of the bag 10 juxtaposed with the periphery 14.
		The closure zone 26 may be extensible in either of two perpendicular directions lying within the plane of the bag 10, although the primary direction of extensibility is generally parallel the fill direction 24.
		One benefit to having a closure zone 26 made of the aforementioned material having two distinct regions is that the ribs of the second region 66 provide an increased tactile sensation and gripping surface for tying together opposed sides of the closure zone 26.
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/98161.011227&ELEMENT_SET=DECL (7331 words)

	ARCC Workshop: Integral Closure, Multiplier Ideals and Cores (Site not responding. Last check: 2007-10-08)
		Loosely speaking, the integral closure of an ideal I is an ideal contained in the radical of I that shares a number of finer properties with I.
		Determining the integral closure of I is a difficult task, which essentially amounts to finding solutions in the ring itself of special polynomial equations whose coefficients belong to higher and higher powers of I.
		The aspects intimately connected to the integral closure that we are planning to focus on are: computation of the integral closure and its complexity; multiplicities and equisingularity theory; cores of ideals and Briancon-Skoda type theorems; multiplier ideals and test ideals; multiplier ideals and jet schemes.
	www.aimath.org /ARCC/workshops/integralclosure.html (442 words)

	Integrally Closed is a Local Property
		Let s be a ring extension of r and let c be the integral closure of r in s.
		If r is integrally closed in s then take the integral closure, which is r, then the fractions by t, and the result is the integral closure of r/t.
		Being integrally closed with respect to s, or with respect to the fraction field of r, is a local property.
	www.mathreference.com /id-ext,icloc.html (636 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A sleeve is threadably mounted on the bottle neck, and an apertured closure carrying a spike is mounted on the sleeve.
		A depending protuberance or spike (9b), integral with the arcuate portion (9a) of the closure (9), is adapted to seal the conventional outlet of the nipple (10) when the sleeve (2) has been turned to the closed position as shown in Fig.
		To further seal the connection between the closure (9) and nipple (10), an annular inwardly extending bead (9c) is formed on the inner surface of the arcuate portion (9a) of the closure (9) and engages the outer surface of the nipple (10).
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=00/47265.000817&ELEMENT_SET=DECL (1327 words)

	Closure Medical -- Recommendations and Resources (Site not responding. Last check: 2007-10-08)
		The algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'', because if ''M'' is any algebraically closed field containing ''K'', then the elements of ''M'' which are algebraic over ''K'' form an algebraic closure of ''K''.
		The algebraic closure of the field of rational numbers is the field of algebraic numbers.
		I phrased the bit about density as a ''fact relating'' it to closure (expressed with "iff") rather than as a ''definition in terms of'' closure (expressed with "if"), since there are alternative definitions of dense sets.
	www.becomingapediatrician.com /health/33/closure-medical.html (1330 words)

	PlanetMath: integrally closed
		is said to be integrally closed (or normal) if it is integrally closed in its fraction field.
		Cross-references: fraction field, integral domain, integral closure, integral, commutative ring, subring
		This is version 11 of integrally closed, born on 2002-04-23, modified 2004-06-18.
	planetmath.org /encyclopedia/IntegrallyClosed.html (94 words)

	Normal scheme - Wikipedia, the free encyclopedia
		In mathematics, in the field of algebraic geometry, a normal scheme is a scheme X for which every stalk (local ring)
		is an integrally closed local ring; that is, each stalk is an integral domain such that its integral closure in its field of fractions is equal to itself.
		An alternate, equivalent, definition uses integral closures in rings of fractions where any nonzero divisor is allowed in the denominator.
	en.wikipedia.org /wiki/Normal_scheme (195 words)

	PlanetMath: integral closure
		It is a theorem that the integral closure of
		This is version 4 of integral closure, born on 2002-01-05, modified 2002-06-09.
		(Commutative rings and algebras :: Ring extensions and related topics :: Integral closure of rings and ideals ; integrally closed rings, related rings)
	planetmath.org /encyclopedia/IntegralClosure.html (81 words)

	Weakly integrally closed domains:\\minimum polynomials of matrices
		The weak integral closure does not always coincide with the set of strongly integral elements, which need not be closed under addition (but is closed under multiplication).
		Of course R is integrally closed, being a unique factorization domain, but the ring R[2i] is not, and R[2i] is weakly integrally closed because it is a free quadratic extension of the Krull domain R.
		Characterize the weakly integrally closed algebras k[M] where M is a submonoid of the free monoid on one generator, and k is a field.
	www.math.fau.edu /richman/Docs/weakly.html (3294 words)

	Department of Mathematics (Site not responding. Last check: 2007-10-08)
		Let I be an ideal in a commutative ring R. Among all the closures of I, the integral closure plays a central role.
		A reduction of I is a subideal having the same integral closure as I. We can think of reductions as simplifications of the given ideal I, which carry most of the information about I itself but, in general, with fewer generators.
		In fact the latter is also the largest ideal integral over I. However, unlike the integral closure, minimal reductions are not unique.
	www.math.unl.edu /print.php?dir=colloquia&file=abstract-20031204.txt (162 words)

	Orders
		This is the integral closure of k[x] in F. EquationOrderInfinite(F) : FldFun -> RngFunOrd
		This is the integral closure of o_(Infinity) in F. IntegralClosure(R, F) : Rng, FldFun -> RngFunOrd
		The integral closure of the subring R of the function field F in itself.
	www.math.niu.edu /help/math/magmahelp/text708.html (2412 words)

	[No title]
		Then the integral closure, H * A , of H in A * is again an unstable algebra over the Steenrod algebra.
		This means that for a ÎA * integral over H , P 1 (a) is integral* over H * and hence over H * and successively P i (a) is integral over H * and hence over H * for all i ÎIN 0.
		However the integral closure H * [n] is nothing else but H * [n] 17.
	www.math.purdue.edu /research/atopology/Neusel/wyoming.txt (1604 words)

	[No title]
		A solution of a differential equation is sometimes called an integral of the equation.
		An element a of a ring B is said to be integral over a ring A contained in B if it is the root of a polynomial with coefficients in A and with leading coefficient 1.
		] The integral closure of a subring A of a ring B is the set of all elements in B that are integral over A.
	www.accessscience.com /Dictionary/I/I13/DictI13.html (2269 words)

	Citebase - Nilpotents, Integral Closure and Equisingularity conditions
		In earlier work, the author described various stratification conditions for a complex analytic set X in terms of the theory of integral closure of modules.
		In this note we show that it is not necessary for X to have a reduced structure to apply the earlier integral closure results.
		As an application we give a simple proof of a necessary and sufficient condition for the Whitney conditions to pass to the intersection of X with a hyperplane containing the smooth stratum Y, and for this intersection to be "typical".
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0507595 (422 words)

	Singular Manual: normalI
		If p is not given, or p==0, compute the closure of all powers up to the maximum degree in t occurring in the generators of the closure of R[It] (so this is the last one that is not just the sum/product of the above ones).
		c is transferred to the procedure primeClosure and toggles its behavior in computing the integral closure of R[It].
		The result is a list containing the closure of the desired powers of I as ideals of the basering.
	www.msri.org /about/computing/docs/singular/sing_604.htm (138 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.
		Let D be an integral domain with quotient field F, and let I be an ideal of D that is invertible when considered as a fractional ideal.
		Let D be an integral domain with quotient field Q, and let F be a finite extension field of Q. If D* is the set of all elements of F that are integral over D, then D* is a Dedekind domain.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	DC MetaData for:Computing the integral closure of an affine semigroup
		DC MetaData for:Computing the integral closure of an affine semigroup
		The authors have developed the computer program "normaliz" for the computation of the Hilbert basis and the Hilbert series of the integral closure of an affine semigroup, where the integral closure can be taken either in the ambient lattice, or within the group of differences (`normalization').
		Affine semigroup rings are the coordinate rings of (not necessarily normal) toric varieties, and homogeneous such rings are the homogeneous coordinate rings of projective toric varieties.
	www.mathematik.uni-osnabrueck.de /preprints/shadow/calg0105.rdf.html (140 words)

	Class Schedule for Class 215 BOTTLES AND JARS
		Encompassing closure removal obstacle and closure engaging for concurrent movement
		Closure removal causes portion thereof to remain with receptacle
		Closure engaging spring arm hinged to receptacle or support thereon
	www.uspto.gov /go/classification/uspc215/sched215.htm (404 words)

	My Research Interests
		The original definition of seminormalization for a ring, R, was the set of elements in the integral closure which were in every R
		In working on cancellation questions, I noticed that there was no theory in the literature outside of the domain case, but there were several occasions where I felt that some element should be an indeterminate because of a transcendence degree argument.
		S and R can be shown to have the same quotient field and one can be in a case where S is integral over R. Showing S is seminormal over R seems critical so I seem to have come full circle or perhaps it is only a matter of using the tools one has.
	www.math.sjsu.edu /~hamann/My_Research_Interests.html (982 words)

	An introduction to field algebra
		Assume that K is a commutative ring with unit, and F is a subring with the same unit.
		Note that if F is a field then ``integral over F'' is the same as ``algebraic over F''.
		Another consequence is that if x is integral over F, and y is integral over F[x], then y is integral over F (again using F[x,y]).
	www.math.harvard.edu /~elkies/M55a.05/galois.html (1508 words)

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