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Topic: Integrally closed

	INTEGRALLY CLOSED
		An equivalent definiton 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 terminology is justified by the fact that the integral closure of R in S is always integrally closed in S, and is in fact the smallest subring of S that contains R and is integrally closed in S''.
		The integral closure of Z in the complex numbers C is the set of all algebraic integers.
	www.websters-online-dictionary.org /definition/english/in/INTEGRALLY+CLOSED.html (355 words)

	Integral closure - Wikipedia, the free encyclopedia
		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).
		In the special case where S is the fraction field of R, the integral closure of R in S is named simply the integral closure of R, and if R is integrally closed in S, then R is said to be integrally closed.
		The integral closure of Z in the complex numbers C is the set of all algebraic integers.
	en.wikipedia.org /wiki/Integral_closure (381 words)

	Integrally closed - Wikipedia, the free encyclopedia
		Though for a lattice-ordered group to be integrally closed and to be archimedean is equivalent.
		We have the surprising theorem that every integrally closed directed group is already abelian.
		This have to do with the fact that an directed group is embeddable into a complete lattice-ordered group iff it is integrally closed.
	en.wikipedia.org /wiki/Integrally_closed (128 words)

	Integrally Closed (Site not responding. Last check: 2007-10-18)
		The integral closure of a ring r in an extension s is the set of elements of s that are integral over r.
		Use corollary 5 to show the integral closure of r in s is integrally closed in s.
		For example, the integers are closed in the rationals.
	www.mathreference.com /id-ext,closed.html (150 words)

	Closure for a container - US Patent 6691901 (Site not responding. Last check: 2007-10-18)
		The cover of claim 33 wherein the shaker flap is retained in the closed position by the wedging interaction of the tab against the inner edge of the at least one shaker opening.
		The spooning flap is retained in the closed position by the wedging interaction of the tab against the inner edge of the spooning opening.
		As the shaker flap 20 is moved into the closed position, cam 39 of lower portion 38 disengages from edge 23, as tab 34 remains engaged with (remains wedged against) edge 23.
	www.patentstorm.us /patents/6691901.html (5406 words)

	[No title]
		It is shown that a semiprime integrally closed Goldie ring is the direct product of a semisimple artinian ring and a finite number of integrally closed invariant domains that are classically integrally closed in their (division) rings of fractions.
		It is shown also that an integrally closed ring has a classical ring of fractions and is classically integrally closed in it.
		It is shown that an integrally closed noetherian ring all of whose nonzero prime ideals are maximal is either a quasi-Frobenius ring or a hereditary invariant domain.
	www.turpion.org /php/paper.phtml?journal_id=sm&paper_id=2576 (134 words)

	Integrally closed domains, minimal polynomials, and null ideals of matrices (Site not responding. Last check: 2007-10-18)
		Abstract: We show that every element of the integral closure D' of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of J. Brewer and F.
		Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x].
		It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A (consisting of all polynomials f in D[x] with f(A)=0) is a principal ideal of D[x].
	blah.math.tu-graz.ac.at /~frisch/icmpabs.html (150 words)

	United States Patent Application: 0040099559
		A flexible element (40) formed as a bent tongue is provided integrally with the bottom or cover for securing plates received within the receptacle (10) against movement in longitudinal direction.
		[0012] Since the flexible element is configured integral with the bottom or cover, it can be prepared together with the bottom or cover in a single manufacturing step, such as by molding, thus avoiding a separate manufacture and a mounting operation within the container.
		When the cover 14 is closed, this resting surface 41 rests resiliently against the front surface of plates 56 received within the container 12, thereby securing those against a displacement in longitudinal direction.
	appft1.uspto.gov /netacgi/nph-Parser?Sect1=PTO1&Sect2=HITOFF&d=PG01&p=1&u=/netahtml/PTO/srchnum.html&r=1&f=G&l=50&s1="20040099559".PGNR.&OS=DN/20040099559&RS=DN/20040099559 (2178 words)

	Patent 4711990: Ceramic heater
		A ceramic heater according to claim 2, wherein said closed loop portion is in the form of a track-like ellipse and said two connecting portions are positioned on a short radius line.
		This ceramic heater is characterized by having a construction in which the closed loop portion and the two electrode portions conjointly form a triangle.
		The heat generating portion comprises a ring-like portion 1 and two leg portions 2 and 2a which are integrally connected to the ring-like portion 1 at two connecting portions 1c and 1d which approximately bisect the ring-like portion 1.
	www.freepatentsonline.com /4711990.html (2673 words)

	ARCC Workshop: Integral Closure, Multiplier Ideals and Cores (Site not responding. Last check: 2007-10-18)
		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 (387 words)

	Algebraic and integral closures
		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.
		It turns out that if A is integrally closed in its own fraction field F, then any A-integral element u of an extension field of F has a minimal polynomial in A[X], and thus satisfies a monic equation over A of degree [F(u):F].
	www.math.harvard.edu /~elkies/M250.04/closure.html (614 words)

	Patent 4129212: Carrier and storage binder for fabric samples
		Structural rigidity of the carrier in a closed configuration with the cover elements overlying and superimposed on the stack of fabric samples is achieved through inclusion of suitable cooperative fastening elements that are attached to opposed surface portions of the header and inner surface of the cover.
		Specifically, the structure in a closed configuration resembles a notebook binder having front and back covers interconnected by a rigid spine that is relatively hinged to both the front and back covers.
		In this closed configuration, the respective headers 41 and 42 are disposed in adjacent relationship with respect to the flanges 49.
	www.freepatentsonline.com /4129212.html (4531 words)

	Integrally Closed is a Local Property
		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.
		This seems to make things worse, because you have to prove the extension is integrally closed for each and every prime p; but in practice you only need examine a few primes, as determined by the polynomial that defines the integral extension.
	www.mathreference.com /id-ext,icloc.html (636 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 12 of integrally closed, born on 2002-04-23, modified 2007-07-27.
	planetmath.org /encyclopedia/IntegrallyClosed.html (94 words)

	Glossary of Terms
		Latch: A moveable, usually spring-loaded pin or bolt, which is part of a lock mechanism, and engages a socket or clip on a door jamb, retaining the door closed.
		Lite: An assembly of glass and a surrounding frame, which is assembled to a door, or is integrally built into the door at the factory.
		Panic-proof Lock: A lock and latch device which permits a door to be opened outward by pressure being applied to a bar mounted across the inside face of the door.
	www.thermatru.com /GlossaryOfTerms.aspx (3799 words)

	commalg.org - the center for commutative algebra
		Given integrally closed $\frak m$-primary ideals $I\supset J$, we show that there is a composition series between $I$ and $J$, by integrally closed ideals only.
		Joseph Lipman and Keiichi Watanabe have posted a new preprint, "Integrally closed finite-colength ideals in two-dimensional regular local rings are multiplier ideals", to the arXiv.
		These are integrally closed ideals with properties that lend themselves to highly interesting applications.
	www.commalg.org /preprints/2002_12.shtml (2168 words)

	Weather strip for vehicle - US Patent 6012760 (Site not responding. Last check: 2007-10-18)
		The weather strip according to claim 1 further comprising a comer portion that is shaped to correspond with a comer of the door, wherein the second section is formed at the comer portion.
		The weather strip according to claim 1, wherein the base of the first section is a generally flat strip of material, wherein the seal is formed along and extends integrally from an edge of the base.
		However, the clip may be integrally formed with the second section 34 by injection molding of TPE.
	www.patentstorm.us /patents/6012760.html (6167 words)

	Weakly integrally closed domains:\\minimum polynomials of matrices
		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.
		It is not inherited by integral quadratic extensions:
	www.math.fau.edu /richman/docs/weakly.html (3294 words)

	Volume 44, number 1 (2000) (Site not responding. Last check: 2007-10-18)
		It is proved that this class of domains coincides with the previous class when R is integrally closed.
		Moreover, these domains are characterized in terms of the altitude formula in case R is not integrally closed.
		An example of a maximal non-universally catenarian subring of its quotient field which is not integrally closed is given (Example 4.2).
	mat.uab.es /~pubmat/v44(1)/44100_05.html (147 words)

	PlanetMath: partially ordered group
		In fact, an directed integrally closed group is an Abelian po-group.
		po-group, l-group, Archimedean po-group, integrally closed po-group, po-semigroup, lattice-ordered group, l-semigroup
		Cross-references: semigroup, axioms, definition, abelian, implies, integrally closed, reals, Archimedean property, archimedean, ordered group, totally ordered, linear order, partial order, lattice, upper bound, filtered set, directed set, identity element, positive, order, preserve, automorphisms, right, sides, inequality, properties, equivalent, poset, group
	planetmath.org /encyclopedia/PartiallyOrderedGroup.html (348 words)

	[No title]
		Theorem 2.4 Suppose that A is integrally closed.
		Since A is integrally closed,* * the localization at any height one prime is a Dedekind domain, and therefore has gl* *obal dimension one.
		Finally, this polynomial sub* ring is integrally* closed, and therefore equal to the invariants.
	hopf.math.purdue.edu /Benson/invdiv.txt (1985 words)

	Dedekind Domains
		is Noetherian, integrally closed in its field of fractions, and the two prime ideals are maximal.
		However, it is not a Dedekind domain because it is not an integral domain.
		is integrally closed, and by Proposition 5.2.5 it is Noetherian.
	modular.fas.harvard.edu /papers/ant/html/node17.html (819 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Suppose that the associated graded ring G is a Cohen-Macaulay domain and that S=R/I is integrally closed.
		Suppose that I has finite projective dimension, that S=R/I is an integrally closed domain of dimension 2, and that the associated graded ring G is a domain.
		Then the conormal module I/I2 is integrally closed if and only if G is normal.¡Èi$R'T/J% ª, >ó¨"Construction of Integral Closures¡##,¨ We have studied the normality of the conormal modules in terms of components of the Rees algebras of the conormal modules.
	www.math.purdue.edu /~hong/rcm.ppt (358 words)

	Matlis, Eben: Torsion-Free Modules
		More specialized problems of finding all integrally closed D-rings are examined in the last seven chapters, where material covered in the first eight chapters is applied.
		An integral domain is said to be a D-ring if every torsion-free module of finite rank decomposes into a direct sum of modules of rank 1.
		After much investigation, Professor Matlis found that an integrally closed domain is a D-ring if, and only if, it is the intersection of at most two maximal valuation rings.
	www.press.uchicago.edu /cgi-bin/hfs.cgi/00/1465.ctl (199 words)

	NEW - CKP/Solo Safety Mat - The Integrally Monitored Safety Mat from Tapeswitch
		They usually have open contacts when the mat is clear and are closed by the weight of someone standing on the mat.
		This orientation is not suitable for safety circuits and so an additional control unit is necessary to provide the normally closed outputs.
		This is much more convenient than the alternative of having an additional control unit close to the mat so that it can provide the volt-free, normally-closed connections to the ASi I/O module.
	www.tapeswitch.co.uk /Product_News/PR_0302.htm (489 words)

	[No title]
		If R* is a graded integral domain of finite transcendence degree with an unstable Ap-action, then R* is integrally closed and finitely generated* * if and only if R* = Un(F R), the set of unstable elements in the graded field of frac tions of R.
		If R* is a graded unstable integral domain of finite transcendence degree with an unstable Ap-action, then R* is integrally closed and finitely ge* nerated as an algebra if and only if R = Un(F R), the set of unstable elements in the graded field of fractions of R.
		Conversely, assume that R* is noetherian and integrally closed.
	hopf.math.purdue.edu /Wilkerson/newring.txt (6367 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Since $\pi$ is the composition of the successive blowing ups of a finite set ${\cal C}$ of infinitely near points (we will call ${\cal C}$ a constellation) to $Q$, the theorem of Zariski on unique factorization of complete ideals allow to describe all these sandwiched surfaces $Y$ as well as the contractions $Z \rightarrow Y$.
		Let $ Q \in S $ be a closed point and set $ R = {\cal O}_{S,Q}$, $X = Spec R$, $Z = T \times_{S} X$ and $ \pi: Z \rightarrow X$ the induced projective birational morphism.
		Finally, we remark that this is an example of non closed characteristic cone obtained by blowing up only ten points (in a chain).} \end{ex} \begin{thebibliography}{99} \bibitem{cgl} A. Campillo, G. Gonz\'{a}lez-Sprinberg, M. Lejeune-Jalabert: {\em Clusters of infinitely near points.} Math.
	home.imf.au.dk /esn/preprints/045 (2038 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.
		In the setting of the previous theorem, if we assume in addition that R is an integrally closed domain, then a further condition holds, known as ``Going down'': Let R be a subring of the integral domain D, assume that D is an integral extension of R, and that R is an integrally closed domain.
		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)

	The On-Line Encyclopedia of Integer Sequences
		Number of monomial ideals in two variables that are artinian, integrally closed, and of colength n.
		Alternatively, "concave partitions" of n, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do no lie in the Ferrers diagram of the partition, is integrally closed.
		the ideal (x^2,y^2) is not integrally closed, hence the partition
	www.research.att.com /~njas/sequences/A084913 (222 words)

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