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Topic: Monoid ring

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	Ring theory Encyclopedia (Site not responding. Last check: 2007-11-02)
		In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.
		Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.
	www.hallencyclopedia.com /topic/Ring_theory.html (994 words)

	Monoid: Definition and Links by Encyclopedian.com
		For instance, it is perfectly possible to have a monoid in which exist two elements a and b and such that a*b = a holds even though b is not the identity element.
		Such a monoid cannot be embedded in a group, because in the group we could multiply both sides with the inverse of a and would get that b = e, which isn't true.
		It is possible to view categories as generalizations of monoids: the composition of morphism in a category shares all properties of a monoid operation except that not all pairs of morphisms may be composed.
	www.encyclopedian.com /mo/Monoid.html (663 words)

	Ring Theory: Rings, Ideals, Integral Domains, Fields - Numericana
		Ring of polynomials whose coefficients are in a given ring.
		The characteristic of a non-unital ring is defined as the least positive integer p such that a sum of p identical terms always vanishes (if there's no such p, then the ring is said to have zero characteristic).
		The radical Rad(I) of an ideal I is the set of all ring elements which have at least one of their powers in I. The radical of an ideal is an ideal.
	home.att.net /~numericana/answer/rings.htm (1318 words)

	Monoid
		Such a monoid cannot be embedded in a group, because in the group we could multiply both sides with the inverse of a and would get that b = e, which isn't true.
		A monoid (M,) has the cancellation property (or is cancellative) if for all a, b and c in M, ab = ac always implies b = c and ba = c*a always implies b = c.
		It is possible to view categories as generalizations of monoids: the composition of morphism in a category shares all properties of a monoid operation except that not all pairs of morphisms may be composed.
	www.ebroadcast.com.au /lookup/encyclopedia/mo/Monoid.html (626 words)

	Monoid ring
		In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid.
		Let R be a ring and G be a monoid.
		The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G].
	pedia.newsfilter.co.uk /wikipedia/m/mo/monoid_ring.html (269 words)

	Springer Online Reference Works
		-regular rings are characterized by the transitivity of perspectivity in the lattice of finitely-generated submodules of the sum of two copies of the ground ring, and also by the fact that a direct sum can be contracted on a finitely-generated projective module (see [4], [8]).
		Every biregular ring with a unit is isomorphic to the ring of global sections with compact support of a sheaf of simple rings with a unit over a compact totally disconnected Hausdorff space, and any such ring of global sections is biregular (see [2]).
		Regular rings were introduced for the coordinization of continuous geometries, biregular rings in connection with the study of functional representations of rings, and Baer (and Rickart) rings in the study of rings of operators.
	eom.springer.de /R/r080830.htm (1289 words)

	Adjoint functors
		For example, an elementary question in ring theory is how to add a multiplicative identity to a ring that doesn't have one (the definition in this encyclopedia actually assumes one: see ring (mathematics) and glossary of ring theory).
		This is the impredicative method: meaning that the ring we are trying to construct is one of the rings quantified over in 'all rings'.
		Similarly, the group ring construction yields a functor from groups to rings, left adjoint to the functor that assigns to a given ring its group of units.
	www.brainyencyclopedia.com /encyclopedia/a/ad/adjoint_functors.html (3503 words)

	Rings (Site not responding. Last check: 2007-11-02)
		Rings are an extension of groups, and you should be familiar with normal subgroups and group homomorphisms in order to proceed with rings.
		in fact a ring is a group and a monoid acting on the same set cooperatively.
		A ring is a set of elements S and two operators + and *.
	www.mathreference.com /ring,intro.html (280 words)

	GAP Manual: 5 Rings
		Because rings are a category of domains all functions applicable to domains are also applicable to rings (see chapter Domains).
		may return the ring of integers of the smallest cyclotomic field in which the elements lie, which need not be the smallest ring overall, because the elements may in fact lie in a smaller number field which is not a cyclotomic field.
		The standard associate of an ring element r of R is an associated element of r which is, in a ring dependent way, distinguished among the set of associates of r.
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP005.htm (3284 words)

	Ring - Eua4xiacwiki
		An abelian ring or commutative ring is a ring whose multiplication (second binary operation) is commutative.
		In a ring not all elements are invertible with respect to multiplication; we denote with Invelm(R) the set of multiplicative invertible elements of R.
		Every field is a commutative ring, or more precisely, forgetting the multiplicative inversion of a field one obtains a commutative ring.
	alice.iac.rm.cnr.it:8080 /wiki/index.php/Ring (248 words)

	Monoid ring (Site not responding. Last check: 2007-11-02)
		In abstract algebra, a monoid ring is a procedure which constructs a new ring algebra ring from a given ring and a monoid.
		The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G ; it is denoted by R [ G ].
		Given a ring R and the monoid of the non-negative integers, N viewed multiplicatively, we obtain the ring R =: R [ x ] of polynomials over that ring.
	www.uk.fraquisanto.net /Monoid_ring (457 words)

	[No title]
		The symmetric monoidal transformation for H is obtained* *, by the universal property of ^, from the map (HA)(n+) ^(HB)(m+) ---!
		This* * con- struction of the monoid ring over S is left adjoint to the functor which takes * a Gamma-ring R to the simplicial monoid* R(1+).
		Since L(HB)* * is naturally isomorphic to B as a simplicial ring, H and L in particular pass to a* *djoint functors between the categories of simplicial B-modules and HB-modules.
	hopf.math.purdue.edu /Schwede/Gamma_algebra.txt (13406 words)

	Monoid ring (Site not responding. Last check: 2007-11-02)
		In abstract algebra a monoid ring is a procedure which constructs a ring from a given ring and a monoid.
		The set of all these functions with these two operations forms a ring monoid ring of R over G ; it is denoted by R [ G ].
		The ring R can be embedded into the ring R [ G ] via the ring homomorphism T : R -> R [ G ] defined by
	www.freeglossary.com /Semigroup_ring (386 words)

	Birgit Reinert's Thesis
		A Gröbner basis G is a set of polynomials such that every polynomial in the polynomial ring has a unique normal form with respect to reduction using the polynomials in G as rules (especially the polynomials in the ideal generated by G reduce to zero using G).
		We prove that the word problem for semi-Thue systems is equivalent to a restricted version of the membership problem for free monoid rings and similarly that the word problem for group presentations is equivalent to a restricted version of the membership problem for free group rings.
		Since monoid rings in general are not commutative, we are mainly interested in right ideals.
	www-madlener.informatik.uni-kl.de /staff/reinert/Privat/diss.long_en.html (1332 words)

	Monoid ring - Encyclopedia, History, Geography and Biography
		Monoid ring - Encyclopedia, History, Geography and Biography
		Let R be a ring and G be a monoid.We can look at all the functions φ : G -> R such that theset {g: φ(g) ≠ 0} is finite.
		φ(k)ψ(l).The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G].If G is a group, then it is called the group ring of R over G.
	www.arikah.com /encyclopedia/Monoid_ring (319 words)

	Vector Enumeration (Site not responding. Last check: 2007-11-02)
		For a ring R, let M be an R-module M, generated as an R-module by s elements {m_1,..., m_s}.
		Suppose there is another ring S, equipped with a ring homomorphism phi: S -> R, such that phi(S) is central in R. In this situation any R-module can be described as an S-module, on which R acts as a ring of S-module endomorphisms.
		Given a finite set X, and a ring S, we can define the free S-algebra A generated by X. This can be seen either as the monoid algebra of the free monoid of words in X, or as all expressions in X and k, combined by addition and multiplication.
	www.umich.edu /~gpcc/scs/magma/text939.htm (1542 words)

	Monoid ring - ExampleProblems.com
		In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid.
		The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G].
		If G is a group, then it is called the group ring of R over G.
	www.exampleproblems.com /wiki/index.php/Monoid_ring (265 words)

	Algebraic categories
		In classical Algebra textbooks, the definition of a ring is more general (the multiplication may not have a neutral element, i.e.
		are examples of commutative rings whereas rings of matrices are not commutative.
		In a commutative ring an element with inverse (right and left) is called a unit and its inverse is called its reciprocal.
	www.csd.uwo.ca /~moreno/CS874/Lectures/Introduction.html/node45.html (787 words)

	The Steinberg group of a monoid ring, nilpotence, and algorithms, by Joseph Gubeladze (Site not responding. Last check: 2007-11-02)
		The Steinberg group of a monoid ring, nilpotence, and algorithms, by Joseph Gubeladze
		For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M].
		This strengthens the K_2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K_2-group is substituted by the unstable ones.
	www.math.uiuc.edu /K-theory/0768 (133 words)

	1. Introduction
		The usual polynomial ring can be viewed as a monoid ring where the monoid is a finitely generated free commutative monoid.
		Another class of non-commutative rings where the elements can be represented by the usual polynomials and which allow the construction of finite Gröbner bases for arbitrary ideals are the so-called solvable rings, a class intermediate between commutative and general non-commutative polynomial rings.
		In case the original reduction ring is a principal ideal ring, only special sets, namely of size two for the G-bases and of size one for the syzygy bases, have to be considered.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/16/paper_html/node1.html (3481 words)

	Counting Functions: Braids LSU REU 2001
		On the other hand, this intersection is the kernel of the natural morphism from the positive braid monoid to the symmetric group, hence a significant fraction $\frac{1}{n!}$ of positive braids are pure.
		It would be interesting to find a nice generating set and study the relationship of this submonoid to the positive braid monoid.
		Artin demonstrated that the braid group on $n$ strands, denoted $B_n$, is a subgroup of the automorphism group of the free group, $F_n$.
	www.math.lsu.edu /~reu/project_braid.htm (653 words)

	2. Fundamental Relations Between Semi-Thue Systems and Monoid Rings
		The ideal membership problem for free group rings with one generator is solvable as this ring corresponds to the ring of Laurent polynomials for the (commutative) free group with one generator.
		Sq87] there are finitely presented monoids with solvable word problem which have no finite convergent presentations and his examples give rise to finitely generated ideals in free monoid rings with solvable ideal membership problem which have no finite Gröbner bases.
		In theorem 2.2 we have shown that the word problem for group presentations is reducible to a restricted version of the ideal membership problem for a free group ring.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/16/paper_html/node2.html (1027 words)

	Algebraic K-theory of monoid rings, by Joseph Gubeladze (Site not responding. Last check: 2007-11-02)
		Algebraic K-theory of monoid rings, by Joseph Gubeladze
		The progress made during last years in the direction mentioned in the title is described.
		More precisely, `classical' theorems concerning stable and non-stable homotopic behavior of algebraic K-functors (starting point of which are Grothendieck-Serre's theorem on K_0-regularity of a regular ring and Quillen-Suslin's solution of Serre's problem) were generalized (including higher K-functors) to the monoid ring extensions corresponding to commutative cancellative monoids.
	www.math.uiuc.edu /K-theory/0068 (106 words)

	Papers
		Gubeladze, Algebraic K-theory of monoid rings, Cloppenburg: Runge.
		Gubeladze, Geometric and algebraic representations of commutative cancellative monoids, Proc A. Razmadze Math.
		Gubeladze, The Steinberg group of a monoid ring, nilpotence, an algorithms, J. Algebra, 307 (2007), 461 - 496.
	math.sfsu.edu /gubeladze/publications.html (494 words)

	Monoid ring (Site not responding. Last check: 2007-11-02)
		The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G ; it is denoted by R''[''G ].
		The ring R can be embedded into the ring R''[''G ] via the ring homomorphism T : R -> R''[''G ] defined by : T''(''r)(1
		A swedish coin with nominal value 50 öre, and the smallest coin in use in Sweden.
	www.serebella.com /encyclopedia/article-Monoid_ring.html (1251 words)

	Semiring - Eua4xiacwiki (Site not responding. Last check: 2007-11-02)
		A semiring is an algebraic structure that can be defined as a set S equipped with two binary operations, that here we call addition and multiplication and denote with the usual symbols + and ·, such that
		(S,·) is a monoid with identity element 1,
		Every ring is a semiring, or more precisely, forgetting its group inversion a ring yields a semiring.
	alice.iac.rm.cnr.it:8080 /wiki/index.php/Semiring (239 words)

	Ring -- the class of all rings
		A ring is a set together with operations +, -, * and elements 0, 1 satisfying the usual rules.
		In this system, it is also understood to be a ZZ-algebra, which means that the operations where one argument is an integer are also provided.
		Each ring is also a member of class Type.
	www.msri.org /about/computing/docs/macaulay/2-0.9/0083.html (206 words)

	MATH 4020
		But we will start with an extended review session on monoids, groups and rings, in which we will touch upon some themes that are not covered in Durbin's book, such as free structures.
		Problem 1: Let R be a commutative unital ring, and let S be a multiplicative submonoid of R. Define an equivalence relation ~ on R x S by (a,s)~(b,t) if there is r in S with rat = rbs.
		For commutative and unital rings R and S, let phi: R--> S be a unital ring homomorphism, and let a, b be in S. Show that there is a unique unital ring homom.
	www.math.yorku.ca /~tholen/math_4020.htm (1268 words)

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