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Topic: Endomorphism ring

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Epimorphism

Monomorphism

	Endomorphism - Wikipedia, the free encyclopedia
		In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself.
		In many but not all situations it is possible to add endomorphisms, and the endomorphisms of a given object then form a ring, called the endomorphism ring of the object.
		An endomorphism that is also an isomorphism is termed an automorphism.
	en.wikipedia.org /wiki/Endomorphisms (232 words)

	Endomorphism ring - Wikipedia, the free encyclopedia
		In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object.
		As the name suggests, the elements of the endomorphism ring of A are the endomorphisms of A, i.e.
		In general, endomorphism rings can be defined for the objects of any preadditive category.
	en.wikipedia.org /wiki/Endomorphism_ring (301 words)

	Automorphism - Wikipedia, the free encyclopedia
		In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.
		In linear algebra, an endomorphism of a vector space V is a linear operator V → V.
		A field automorphism is a bijective ring homomorphism from a field to itself.
	www.wikipedia.org /wiki/Automorphism (898 words)

	Rings (Site not responding. Last check: 2007-09-18)
		The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers.
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
		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.risberg.ws /Hypertextbooks/Mathematics/Algebra/rings.htm (890 words)

	Endomorphism ring -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-18)
		As the name suggests, the elements of the endomorphism ring of A are the (Click link for more info and facts about endomorphism) endomorphisms of A, i.e.
		We can define the endomorphism ring of any (A self-contained component (unit or item) that is used in combination with other components) module in exactly the same way; instead of group homomorphisms we then take module homomorphisms of course.
		In general, endomorphism rings can be defined for the objects of any (Click link for more info and facts about preadditive category) preadditive category.
	www.absoluteastronomy.com /encyclopedia/e/en/endomorphism_ring.htm (471 words)

	division ring (Site not responding. Last check: 2007-09-18)
		In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e.
		Division rings are very similar to fields except that their multiplication is not required to be commutative.
		In general, if R is a ring and S is a simple module over R, then the endomorphism ring of S is a division ring; every division ring arises in this fashion from some simple module.
	en.mcfly.org /division_ring (206 words)

	Local ring
		In abstract algebra, local rings are certain rings that are comparatively simple and serve to describe the local behavior of functions defined on varieties or manifolds.
		The exact same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point.
		The Jacobson radical m of a local ring R (which is equal to the unique left maximal ideal and also to the unique right maximal ideal) consists precisely of the non-units of the ring; furthermore it is the unique two-sided maximal ideal of R.
	www.teachersparadise.com /ency/en/wikipedia/l/lo/local_ring.html (1015 words)

	Learn more about Ring (mathematics) in the online encyclopedia. (Site not responding. Last check: 2007-09-18)
		Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are called associative rings.
		For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations.
		Given a ring R and an ideal I of R, the factor ring R/I is the set of cosets of I together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
	www.onlineencyclopedia.org /r/ri/ring__mathematics_.html (835 words)

	The Ring of Endomorphisms (Site not responding. Last check: 2007-09-18)
		Thus s is a subring of the endomorphism ring of m.
		When r is a ring, it acts as a left r module, which admits a right r module s, namely the r endomorphisms of r.
		This is a ring endomorphism that fixes 1, but there is no left r module endomorphism that fixes 1, other than the identity map.
	www.mathreference.com /mod,rend.html (517 words)

	Simple module - Wikipedia, the free encyclopedia
		In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S.
		Understanding the simple modules over a ring is usually helpful because they form the "building blocks" of all other modules in a certain sense.
		The converse of Schur's lemma is not true in general: there are non-simple modules whose endomorphism ring is a division ring.
	www.encyclopedia-online.info /Simple_module (326 words)

	Phill Schultz LIST OF PUBLICATIONS
		The endomorphism ring of the additive group of a ring, Journal Aust.
		The Jacobson radical of the endomorphism ring of a valued vector space, [with L. Fuchs], in Algebraic Structures and Applications, Lecture Notes in Pure and Applied Mathematics, Vol.
		The endomorphism ring of a bounded abelian p-group [with M. Avino-Diaz], in Abelian groups, rings and modules, Eds.
	www.maths.uwa.edu.au /~schultz/publications.html (787 words)

	Endomorphism ring (Site not responding. Last check: 2007-09-18)
		As the suggests the elements of the endomorphism ring A are the endomorphisms of A i.e.
		We can define the endomorphism ring of module in exactly the same way; instead group homomorphisms we then take module homomorphisms course.
		The Lord of the Rings: The Fellowship of the Ring
	www.freeglossary.com /Endomorphism_ring (481 words)

	PlanetMath: ring of endomorphisms (Site not responding. Last check: 2007-09-18)
		Instead of writing endomorphisms as functions, it is often convenient to write them multiplicatively: we simply write the application of the endomorphism
		There is a complication, however, if we still think of endomorphism as functions written on the left.
		This is version 5 of ring of endomorphisms, born on 2003-11-25, modified 2005-06-29.
	planetmath.org /encyclopedia/RingOfEndomorphisms.html (305 words)

	[No title] (Site not responding. Last check: 2007-09-18)
		ring2 http://www.openmath.org/cd/ring2.ocd 2006-06-01 2004-06-01 1 1 experimental Basic functions for homomorphisms in ring theory Initiated by Arjeh M. Cohen 2004-02-25 is_homomorphism This symbol is a boolean function with three arguments.
		right_multiplication This symbol is a function with two arguments, which should be a ring M and an element x of M. When applied to M and x, it denotes right multiplication on M by x.
		are_isomorphic This symbol is a boolean function with two arguments, which are rings M, N. When applied to M and N, it denotes the fact that there is an isomorphism from M to N.
	www.win.tue.nl /~amc/oz/om/cds/ring2.html (109 words)

	HJM, Vol. 29, No. 3, 2003
		If R is a unital ring, then the left multiplications by elements of R obviously form endomorphisms of the additive group of R. In fact they form a group direct summand of the endomorphism ring and if the complement is trivial, then these rings are called E-rings which are well-studied.
		Finally, we show that in ZFC there exists an almost-free ring R of minimal uncountable cardinality such that the endomorphism ring of R is isomorphic to the direct sum of the integers and R itself.
		The weak Baer going-down (resp.,universally going-down) rings are characterized as the EGD (resp., EUGD) rings whose total quotient ring is von Neumann regular.
	www.math.uh.edu /~hjm/Vol29-3.html (2232 words)

	Ring (mathematics) at opensource encyclopedia (Site not responding. Last check: 2007-09-18)
		In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
		Given a ring R and an ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations
		Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring.
	wiki.tatet.com /Ring_(mathematics).html (933 words)

	MTH-3E30 : Ring Theory (Site not responding. Last check: 2007-09-18)
		Overview: Ring theory has many applications in mathematics and thanks to the re-discovery of its connections to physics in the last fifteen years is playing an ever more important role.
		The idea of a module and its endomorphism ring is crucial for the theory.
		One of the most powerful and efficient methods in non-commutative ring theory is the analysis of their representations.
	www.mth.uea.ac.uk /maths/syllabuses99-00/3E3001.html (335 words)

	commalg.org - the center for commutative algebra
		Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent.
		Abstract: This is a survey on the relation between homological properties of the Frobenius endomorphism and finiteness of various homological dimensions of the ring or of modules over it, such as global dimension and projective dimension.
		We begin with Kunz's surprising result in 1969 that the regularity of a Noetherian local ring is equivalent to the flatness of its Frobenius endomorphism, as well as the subsequent generalizations to the module setting by Peskine and Szpiro and continue up through the recent flurry of results in the last five years.
	www.commalg.org /preprints/2003_01.shtml (2272 words)

	Endomorphism - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-09-18)
		So, for example, an endomorphism of a vector space V is a linear map f : V → V and an endomorphism of a group G is a group homomorphism f : G → G, etc. In general, we can talk about endomorphisms in any category.
		Endomorphism Rings of Abelian Groups (Algebras and Applications)
		Module Theory : Endomorphism rings and direct sum decompositions in some classes of modules (Progress in Mathematics)
	encyclopedia.worldsearch.com /endomorphism.htm (300 words)

	Dr. R. N. Gupta, Professor (Site not responding. Last check: 2007-09-18)
		Written a paper 'Nil semigroups of endomorphism rings of modules with chain condition'(jointly with Dinesh Khurana) in which we have proved that: If M is a tau torsionfree, tau Artinian (tau torsionfree, tau Noetherian) module then nil multiplicative closed subsets of End(M) are nilpotent and the indices of nilpotency are bounded.
		Zelmanowitz, J. [ On the endomorphism ring of a discrete module: a theorem of Kasch, Advances in Ring Theory, eds Jain, S. and Rizvi, S. T., Birkhauser, 1997, 31-322] proved that the endomorphism ring of a discrete module modulo its Jacobson radical is isomorphic to a direct product of full linear rings.
		On the existence of Koethe radical and its coincidence with lower nil radical for a class of rings, Niew Archief Voor Wiscunde (3), 16 (1968), 90-93, MR 39#6917.
	www.puchd.ac.in /maths/rng.html (1107 words)

	ABSTRACT ALGEBRA ON LINE: Structure of Noncommutative Rings (Site not responding. Last check: 2007-09-18)
		P, for any ideals A, B of R. A proper ideal I of the ring R is called a semiprime ideal if it is an intersection of prime ideals of R. A proper ideal P of the ring R is called a left primitive ideal if it is the annihilator of a simple left R-module.
		The ring R is called a simple ring if (0) is a maximal ideal; it is called a prime ring if (0) is a prime ideal, and a semiprime ring if (0) is a semiprime ideal.
		If P is a primitive ideal of the ring R, then there exists a division ring D and a vector space V over D for which R/P is isomorphic to a subring of the ring of all linear transformations from V into V. Proposition.
	www.math.niu.edu /~beachy/aaol/noncommutative.html (1031 words)

	iqexpand.com (Site not responding. Last check: 2007-09-18)
		The term endomorphism derives from the Greek adverb endon (andquot;insideandquot;) and morphosis (andquot;to formandquot; or andquot;to shapeandquot;).In algebra, an endomorphism of a group, module, ring, vector space, etc. is a homomorphism...
		Endomorphism Graphs Next: Fixed Point Free Automorphisms Up: Complexity Previous: Basic Issues Endomorphism Graphs Recall that a graph Â is a pair G =(V, E) of a set V of vertices Â and a set E of doubleton subsets...
		Iterated Endomorphisms of the Plane The Fractal Rabbit Endomorphism Wing of the Savage Rabbit Art Gallery proudly presents these creations by Savage Rabbit Artist and SCO employee Ronald Joe Record...
	endomorphism.iqexpand.com (468 words)

	[No title]
		R* ecall that the centre of an additive category is the endomorphism* ring of the identit* *y functor.
		For instance, using some different methods, he shows that the endomorphism ring* * of an idempotent module EC is local provided that C corresponds to a connected clo* sed homogeneous subvariety of VG (k) (or to a collection of closed homogeneous subv *arieties of VG (k) which is connected in a suitable sense).
		Given an additive category A, we denote by Z[A] the centre of A which is the endomorphism ring of the identity f* *unctor idA.
	hopf.math.purdue.edu /KrauseH/idempotent.txt (5343 words)

	[No title]
		Recall that in this context, $\sigma$ is a ring endomorphism of $R$, $\delta$ is a $\sigma$-derivation of $R$ (i.e.\ a map $\delta: R\la R$ satisfying $\delta(a+b)=\delta(a)+\delta(b)$ and $\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$ for all $a,b\in R$), and multiplication of polynomials in $R[x;\sigma,\delta]$ (written with left-hand coefficients) is determined by the rule $xr = \sigma(r)x+\delta(r)$ for every $r\in R$.
		Part of the attraction of 2-primal rings (in addition to their being a common generalization of commutative rings and rings without nilpotent elements) lies in the structure of their prime ideals.
		In the investigation of skew polynomial rings, the 2-primal condition was studied via Shin's minimal-prime criterion, based on an understanding of extension and contraction of primes between the base ring the the skew polynomial extension.
	euler.slu.edu /Dept/Faculty/marks/On2PrimalOreExtensions.txt (2911 words)

	[No title]
		One application is that any object X 2 M has an associated `homotopy endomorphism ring spectrum' hEnd (X) (where by ring spectrum we mean essentially what used to be called an A1 -ring spectrum).
		These ring spectra, as well as the overall enrichment by symmetric spectra, are homot* *opy invariants of the model category M. An application.
		Unlike the homotopy endomorphism spectra of Section 5.6, however, this dga is not an invariant of Quillen equivalence.
	hopf.math.purdue.edu /Dugger/spenrich.txt (12833 words)

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