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Topic: Subring

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	PlanetMath: subring
		In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroups in group theory.
		This is version 14 of subring, born on 2002-03-02, modified 2005-12-16.
		The sufficient (and necessary) subring criterion is that the non-empty S is closed under subtraction and multiplication.
	planetmath.org /encyclopedia/Subring.html (191 words)

	Subring - ExampleProblems.com
		In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations.
		More precisely, given a ring (R, +, ), we say that a subset S of R is a subring* of R if it is a ring under the restriction of + and * to S, and contains the same multiplicitive identity as R.
		For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X].
	www.exampleproblems.com /wiki/index.php/Subring (307 words)

	PlanetMath: subring
		is a subring since the difference and product of two even numbers is again an even number.
		This is version 8 of subring, born on 2002-03-02, modified 2004-11-15.
		The sufficient (and necessary) subring criterion is that the non-empty S is closed under subtraction and multiplication.
	www.planetmath.org /encyclopedia/Subring.html (181 words)

	Subring -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-20)
		More precisely, given a ring (R, +, ), we say that a (A set whose members are members of another set; a set contained within another set) subset S of R is a subring* thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R.
		A subring is just a ((mathematics) a subset (that is not empty) of a mathematical group) subgroup of (R, +) which contains the identity and is closed under multiplication.
		The ring Z has subrings of the form nZ, where n is any integer.
	www.absoluteastronomy.com /encyclopedia/s/su/subring.htm (126 words)

	PlanetMath: unities of ring and subring
		Note: In the cases 3 and 4, the unity of the subring
		"unities of ring and subring" is owned by pahio.
		This is version 2 of unities of ring and subring, born on 2004-11-18, modified 2004-11-19.
	www.planetmath.org /encyclopedia/UnitiesOfRingAndSubring.html (110 words)

	Read about Subring at WorldVillage Encyclopedia. Research Subring and learn about Subring here! (Site not responding. Last check: 2007-08-20)
		mathematics, a subring is a part of a
		More precisely, given a ring (R, +, ), we say that a subset S of R is a subring* thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R.
		For example, the ring Z of integers is a subring of the
	encyclopedia.worldvillage.com /s/b/Subring (156 words)

	Subring (Site not responding. Last check: 2007-08-20)
		A subring is just a subgroup of (R, +) which contains the identity and is closed under multiplication.
		For example, the ring 'Z of integer s is a subring of the field of real number s and also a subring of the ring of polynomial s Z'''[''X ].
		Every ring has a unique smallest subring, isomorphic to either the integers Z or some modular arithmetic Z' n''
	www.serebella.com /encyclopedia/article-Subring.html (155 words)

	Station Information - Subring
		A subring is just a subgroup of (R, +) which contains 1 and is closed under multiplication.
		For example, the ring Z of integers is a subring of the field (mathematics) of real numbers and also a subring of the ring of polynomials Z[X].
		Every ring has a unique smallest subring, isomorphic to either the integers Z or some modular arithmetic Z
	www.stationinformation.com /encyclopedia/s/su/subring_1.html (118 words)

	Ring (mathematics) - Wikipedia, the free encyclopedia
		The split-complex plane D is a ring useful in modern physics and is a subring of the tessarines.
		If a subset S of a ring R is itself a ring with the same operations (restricted to S), and the identity element 1 of R is contained in S, then S is called a subring of R.
		We say that a subring S of R is central if it is a subring of the center of R.
	en.wikipedia.org /wiki/Ring_(mathematics) (1102 words)

	PlanetMath: generated subring
		Of particular interest is the subring generated by a family of subrings
		Cross-references: complete lattice, generated by, subfield, words, composite field, quotient field, quotients, fields, monomials, sums, finite, subrings, intersection, ring, subset
		This is version 6 of generated subring, born on 2007-04-20, modified 2007-05-03.
	planetmath.org /encyclopedia/GeneratedSubring.html (149 words)

	subring - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "subring" is defined.
		subring : The American Heritage® Dictionary of the English Language [home, info]
		Subring : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?w=subring (118 words)

	Encyclopedia: Ring (mathematics)
		In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication.
		Given a ring (R, +, ), we say that a subset S of R is a subring* thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R. A subring is just a subgroup of (R, +) which contains the identity and is closed under...
		The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements.
	www.nationmaster.com /encyclopedia/Ring-%28mathematics%29 (2731 words)

	Abstract algebra:Rings, fields and modules - Wikibooks
		So, just as we did for groups, we note that if we're considering a subset of something that's already a ring, then our job is easier.
		Definition: A subring S of a ring R is a subset of R that is a ring (under the same two operations as for R).
		is a subring of the complex numbers C.
	www.wikibooks.org /wiki/Abstract_algebra:Rings (317 words)

	[No title]
		Abstract: For certain subrings of the mod-$p$-cohomology of a compact Lie group, we give a description of the spectrum, analogous to Quillen's description of the spectrum of the whole cohomology ring.
		Status: Submitted for publication Abstract: In the cohomology ring of an extraspecial $p$-group, the subring generated by Chern classes and transfers is studied.
		This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical.
	claude.math.wesleyan.edu /~mhovey/archive/letter33 (782 words)

	sharewarebase.de - KPN_Qwest - jetzt wirds heiss (Site not responding. Last check: 2007-08-20)
		Subring A verbindet Hamburg, Bremen, Hannover, Magdeburg und Berlin miteinander.
		Subring B versorgt Kunden in Hannover, D³sseldorf, K÷ln, Bonn, Frankfurt, N³rnberg, Leipzig und Magdeburg.
		Der s³dliche Subring C ³berlappt sich in Frankfurt und Stra▀burg mit Euroring 1 und verbindet Mannheim, Karlsruhe, Stra▀burg, Stuttgart und M³nchen.
	www.shareware-base.de /modules.php?name=News&file=print&sid=2560 (331 words)

	Hurwitz quaternion (Site not responding. Last check: 2007-08-20)
		In mathematics, the Hurwitz quaternions are a subring of the ring H of all quaternions, consisting of all quaternions
		It can be checked that this set of quaternions is closed under quaternionmultiplication, so that it indeed forms a subring.
		It is not amaximal order, and therefore (as it turns out) less suitable for developing a theory of left ideals comparable to that of algebraic number theory.
	www.therfcc.org /hurwitz-quaternion-210846.html (226 words)

	A Graded Subring of an Inverse Limit of Polynomial Rings (ResearchIndex) (Site not responding. Last check: 2007-08-20)
		A Graded Subring of an Inverse Limit of Polynomial Rings (1998)
		8 Grobner bases and normal forms in a subring of the power ser..
		3 Some properties of a subring of the power series ring on a c..
	citeseer.lcs.mit.edu /snellman98graded.html (426 words)

	[No title] (Site not responding. Last check: 2007-08-20)
		A subring is a \htmladdnormallink{subset}{http://planetmath.org/encyclopedia/Set.html} $S$ of $A$ with the operations $+$ and $*$ of $A$ \htmladdnormallink{restricted}{http://planetmath.org/encyclopedia/Quantifier.html} to $S$ and such that $S$ is a ring by itself.
		A subring is called an \htmladdnormallink{ideal}{http://planetmath.org/encyclopedia/Ideal.html} if whenever $s\in S$ and $a\in A$, it happens that $sa\in S$.
		Then $(2\Z,+,\cdot)$ is a subring since the sum or product of two \htmladdnormallink{even numbers}{http://planetmath.org/encyclopedia/EvenNumber.html} is again an even number.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/massive/Subring.tex (103 words)

	[No title] (Site not responding. Last check: 2007-08-20)
		Here's one way to state the theorem: If K is a field, R is a subring of K, and gen_R K is finite, then there exists r in R-{0} such that R[1/r] is a field and len_{R[1/r]} K is finite.
		Here len means length as a module (e.g., dimension as a vector space), and gen means minimum number of generators as an algebra.
		In particular, if K is a field, R is a subring of K, gen_R K is finite, and R is a quotient of a Jacobson ring, then R is a field and len_R K is finite.
	cr.yp.to /zgk.html (643 words)

	Some Properties of a Subring of the Power Series Ring on a Countably Infinite Number of Variables Over a Field - ... (Site not responding. Last check: 2007-08-20)
		K[x 1 ; : : : ; x n ], we endow R 0 with a topology, and show that with respect to this topology, homogeneous, finitely generated ideals are closed.
		A Graded Subring of an Inverse Limit of Polynomial Rings - Snellman (1998)
		Some properties of a subring of the power series ring on a countably infinite number of variables over a field.
	citeseer.lcs.mit.edu /snellman97some.html (640 words)

	Subrings
		This together means that if S1:1 and 2 holds, then will be a subring of .
		A, S1 will thus be true, and A will be a subring.
		An example of a subring of R could be Z[
	hemsidor.torget.se /users/m/mauritz/math/alg/subring.htm (379 words)

	Subring (Site not responding. Last check: 2007-08-20)
		Given a ring (R + ) we say that a subset S of R is a subring* thereof if it is a ring the restriction of + and * thereto contains the same unity as R.
		For example the ring Z of integers is a subring of the field of real numbers and also a subring of the of polynomials Z [ X ].
		Every ring has a unique smallest subring to either the integers Z or some modular arithmetic Z
	www.freeglossary.com /Subring (452 words)

	Articles and preprints (Site not responding. Last check: 2007-08-20)
		Non-homogeneous ideals in a graded subring of the power series ring on a countably infinite number of variables over a field International Journal of Mathematics, Game Theory, and Algebra, volume 10, no 5, 2000, pp 391-404.
		Some topological properties of a subring of the power series ring on a countably infinite number of variables over a field International Journal of Mathematics, Game Theory, and Algebra, volume 8, no 4, 1999.
		Gröbner bases and normal forms in a subring of the power series ring on countably many variables over a field, Journal of Symbolic Computation 1998, vol 25, number 3, pp 315-328.
	www.matematik.su.se /~jans/public/math (530 words)

	On Non-Homogeneous Ideals in a Subring of the Power Series Ring on a Countably Infinite Number of Variables Over a ... (Site not responding. Last check: 2007-08-20)
		On Non-Homogeneous Ideals in a Subring of the Power Series Ring on a Countably Infinite Number of Variables Over a Field (1997)
		We extend a result from [5], namely that locally finitely generated ideals in the ring R 0 are closed, to show that a (non-homogeneous) ideal I in R 0, whose associated homogeneous ideal gr T (I) is locally finitely generated, is closed.
		Introduction In [5], an exhaustive and separated decreasing filtration of ideals was given on the ring R 0, defined as the smallest...
	sherry.ifi.unizh.ch /snellman97nonhomogeneous.html (347 words)

	Does there exist a ring whose subring has a unity but the ri (Site not responding. Last check: 2007-08-20)
		Does there exist a ring whose subring has a unity but the ring it
		Does there exist a ring whose subring has a unity but the ring it self
		elements with support in that set form a subring which has a unity.
	www.newsbackup.com /about678298.html (297 words)

	Ideals (Site not responding. Last check: 2007-08-20)
		A subring of a ring r is a subgroup of r under +, that is closed under *.
		We'll assume the subring contains 1, so it is a proper ring.
		The integers are a subring of the rationals.
	www.mathreference.com /ring,ideal.html (333 words)

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