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Topic: Semisimple ring

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	PlanetMath: semisimple ring
		A ring is left semisimple iff it is semiprimitive and left artinian.
		A ring is left semisimple iff it is von Neumann regular and left noetherian.
		The theorem implies that a left semisimplicity is synonymous with right semisimplicity, so that it is safe to drop the word left or right when referring to semisimple rings.
	planetmath.org /encyclopedia/SemisimpleRing2.html (254 words)

	PlanetMath: Baer ring
		And, by Wedderburn-Artin theorem, a semisimple ring is also a Baer ring.
		Another example, found in operator theory, is the ring of bounded linear operators on a Hilbert space.
		This is version 2 of Baer ring, born on 2006-04-23, modified 2006-04-23.
	planetmath.org /encyclopedia/BaerRing.html (273 words)

	Jacobson Semisimple and Quotient Rings
		A ring r is jacobson semisimple if its jacobson radical is 0.
		Use criteria 4 to assert xy is a unit.
		A simple ring has only one proper ideal, namely 0, and this must be the jacobson radical.
	www.mathreference.com /ring-jr,ss.html (734 words)

	Dinh Van Huynh's homepage :: Publications
		Uber Ringe mit eingeschr ankter Minimalbedingung hoherer Stufe fur Unterringe, (with A. Widiger), Beitr.
		Prime Goldie rings of uniform dimension at least two and with all one-sided ideals CS are semihereditary, (with S.K. Jain, S.R. Lopez-Permouth), Comm.
		Rings characterized by CS condition of their modules, Trends in Rings and Modules, (to appear).
	www.math.ohiou.edu /~huynh/publications.html (825 words)

	UWM Math: Noetherian Rings (Site not responding. Last check: 2007-10-13)
		The main thrust of the theory of commutative rings is intimately related to the theory of rings of polynomial functions (and rings derived from them such as quotients and localizations).
		The study of non-commutative rings is a field begun in the 20th century, and much of the early work concentrated on division rings and algebras that were finite dimensional over a field.
		While many interesting ring theoretic results were proven in between, it is probably fair to say that the modern study of non-commutative noetherian rings began with A. Goldie's work in 1958-1960 giving necessary and sufficient conditions for a ring to have a semisimple ring of fractions.
	www.uwm.edu /Dept/Math/Research/Algebra/noetherian/noetherian.html (467 words)

	Semisimple module - Wikipedia, the free encyclopedia
		Here, the base ring is a ring with unity, though not necessarily commutative.
		A semisimple module may be characterised as being (isomorphic to) a direct sum of simple modules.
		Semisimple rings are also small (they're both Artinian and Noetherian).
	en.wikipedia.org /wiki/Semisimple_module (385 words)

	Ring -- from Wolfram MathWorld
		(a ring satisfying this property is sometimes explicitly termed an associative ring).
		A ring satisfying all additional properties 6-9 is called a field, whereas one satisfying only additional properties 6, 8, and 9 is called a skew field.
		Rings which have been investigated and found to be of interest are usually named after one or more of their investigators.
	mathworld.wolfram.com /Ring.html (616 words)

	Semisimple - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-13)
		A semisimple module is one in which each submodule is a direct summand.
		In particular, a semisimple representation is completely reducible, i.e., is a direct sum of irreducible representations (under a descending chain condition).
		Every finite dimensional representation of a semisimple Lie algebra, Lie group, or algebraic group in characteristic 0 is semisimple, i.e., completely reducible, but the converse is not true.
	en.wikipedia.org.cob-web.org:8888 /wiki/Semisimple (346 words)

	[No title]
		In the derived* * category of a ring, for example, the sphere is the chain complex with R in degree zero a* *nd zero elsewhere.
		Observe that since Proposition 3.6 applies to noncommuative rings, * we may drop the commutativity assumption for one direction of this theorem: GH is tru e in the derived category of (right) modules over a von Neumann regular ring*.
		A ring R is semisimple if, as a rig* *ht module over itself, it decomposes as a finite direct sum of simple modules.
	hopf.math.purdue.edu /Lockridge/gh.txt (5978 words)

	APPENDIX J
		A ring R is a module with an added multiplication operation (not necessarily commutative) where both right handed and left handed distributivity as well as associativity hold.
		A division ring R is a ring wherein the elements in R^* the nonzero elements of R, all have multiplicative inverses.
		A Ring Homomorphism is a mapping from a ring R to a ring R' which preserves the ring operations.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (6145 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Ofer Hadas: "On Smoktunowicz's Construction of a nil ring with non-nil polynomial ring" A counter-example to Amitsur's conjecture "if R is a nil ring then R[X] is nil" was recently constructed by A. Smoktunowicz.
		Yuval Ginosar: "Semisimple Strongly Graded Rings" Any strongly graded ring induces a group homomorphism (a generalized collective character) from the grading group G to the Picard group of the coefficient ring R. In this talk we discuss the problem of the realization of such a character by a strongly graded ring.
		In particular, when the coefficient ring is semisimple we give a criterion to the existence of a semisimple realization.
	imu.org.il /Meeting01/algebra.txt (493 words)

	Group ring - Wikipedia, the free encyclopedia
		In mathematics, a group ring is a ring R[G] constructed from a ring R and a multiplicative group G.
		An example of a group ring of an infinite group is the ring of Laurent polynomials: this is exactly the group ring of the infinite cyclic group Z.
		There is an elegant characterization from category theory of the group ring construction with a fixed ring R as the left adjoint to the functor taking an associative R-algebra with one to its group of units.
	en.wikipedia.org /wiki/Group_ring (850 words)

	Artin Wedderburn Theorem
		In the above, we started with a semisimple ring r and built a block b, an ideal in r, which is a ring, and a semisimple r module, and a b module.
		Combine this with addition, and the ring of matrices is isomorphic to the ring of endomorphisms of r, which is isomorphic to r.
		A semisimple ring, which is both left and right semisimple after all, is left and right artinian and noetherian.
	www.mathreference.com /mod-simp,sring.html (2771 words)

	philosophy notes
		To say that a group (ring) is semisimple is to say that a certain normal subgroup (ideal) is trivial.
		Every semisimple ring with the descending chain condition on left ideals is completely reducible.
		The fact that a polynomial ring over a field is Noetherian yields to a descending chain condition on closed subgroups.
	www.math.uic.edu /~jbaldwin/pub/phil.html (2052 words)

	Final Exercises
		Show that if a ring R contains a non-zero nilpotent left ideal then R is not semisimple.
		Show that if the ring R is Artinian and commutative then J(R) is the set of nilpotent elements of R.
		Let M be a semisimple left R-module for a ring R.
	www.maths.warwick.ac.uk /~rumynin/rings2002/ln/node29.html (325 words)

	Construction of Lie Algebras
		A sequence of n^3 elements of the ring R. The sequence elements are the structure constants themselves, in the order a_(11)^1, a_(11)^2,..., a_(11)^n, a_(12)^1, a_(12)^2,..., a_(nn)^n.
		The twisted semisimple Lie algebra over the finite field k with Cartan type N given as a string and twist given by the permutation p.
		The semisimple Lie algebra with crystallographic Cartan matrix C over the ring k (see Section Cartan Matrices).
	www.math.lsu.edu /magma/text1062.htm (2092 words)

	Von Neumann regular ring - Wikipedia, the free encyclopedia
		The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular.
		Generalizing the above example, suppose S is some ring and M is an S-module such that every submodule of M is a direct summand of M (such modules M are called semisimple).
		A ring is semisimple artinian if and only if it is von Neumann regular and left (or right) Noetherian.
	en.wikipedia.org /wiki/Von_Neumann_regular_ring (346 words)

	Semisimple and DCC
		A ring r is left semisimple iff it is jacobson semisimple and it exhibits dcc on its principle left ideals.
		Combine this with the fact that a left semisimple ring is left artinian, and the principle left ideals are dcc.
		Now r is left artinian iff it is left semisimple iff it is right semisimple iff it is right artinian.
	www.mathreference.com /ring-jr,jdcc.html (466 words)

	Citebase - Local morphisms and modules with a semilocal endomorphism ring (Site not responding. Last check: 2007-10-13)
		An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings.
		A ring homomorphism is said to be local if it carries non-units to non-units.
		Semilocal rings can be characterized as those rings having a local homomorphism to a semisimple artinian ring.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0510104 (233 words)

	Operations on Lie Algebras
		The map giving the morphism from the (structure constant) Lie algebra L to M. Either L is a subalgebra of M, in which case the embedding of L into M is returned, or M is a quotient algebra of L, in which case the natural epimorphism from L onto M is returned.
		The coefficient ring (or base ring) over which the Lie algebra L is defined.
		Given an algebra L and either a subalgebra S of dimension m of L or a sequence Q of m linearly independent elements of L, return a sequence containing a basis of L such that the first m elements are the basis of S resp.
	www.math.lsu.edu /magma/text1065.htm (1072 words)

	OhioLINK ETD: Roman, Cosmin
		A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property.
		A ring over which every module is Baer is shown to be precisely a semisimple Artinian ring.
		Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general.
	www.ohiolink.edu /etd/view.cgi?osu1092676774 (295 words)

	[No title]
		For an associative ring R, let A be the category of left R-modules, let P be the collection of all summands of free R-modules and let E be the collection of all surjections of R-modules.
		S is a map of rings that is split monic as a map of R-modules, then S is difficult.
		On the other hand, if R is a semisimple ring, or equivalently, if every R-module is projective, then the trivial projective class is the same as the categorical projective class, and so is determined by a set (the set {R}).
	jdc.math.uwo.ca /papers/relative.txt (10317 words)

	Fellowship of the Ring Seminar
		I'll then introduce a generalization of this classification, to rings resembling the ring of power series (over a p-adic field) convergent in an annulus; it bears a formal resemblance to the classification of vector bundles on the projective line, including the presence of a complete numerical invariant that looks like a Harder-Narasimhan polygon.
		Abstract: The small quantum group Uf was introduced by Lusztig as a certain finite dimensional Hopf algebra associated to a semisimple complex Lie algebra Lg and a primitive l-th root of unity q in C. According to Lyubashenko and Majid, in many cases Uf admits a bijective action of two operators obeying the modular identities.
		Abstract: I shall describe the notion of a slightly ramified extension of local rings, and use this to prove a criterion for the smoothness of deformation spaces (or local moduli spaces) in mixed characteristic in terms of lifting the tangent space of the problem.
	wpad.brandeis.edu /~fdiamond/pastfor.html (6881 words)

	[ref] 61 Lie Algebras
		is a semisimple subalgebra complementary to the radical of
		Currently this is only implemented for semisimple associative algebras, and Lie algebras (semisimple or not).
		Let L be a semisimple Lie algebra over a field of characteristic 0, and let R be its root system.
	www.gap-system.org /Manuals/doc/htm/ref/CHAP061.htm (4974 words)

	Rings and Modules, MAS427 (Site not responding. Last check: 2007-10-13)
		In this way the object generates a ring R, and we can look at the family of all abelian groups on which this ring R acts; these groups are the R-modules.
		Structure theorems: chain conditions on rings and modules, Noetherian rings, Artinian rings, Artin-Wedderburn Theorem and the structure of finitely generated modules over principal ideal domains.
		equivalence of the characterisations of a semisimple ring,
	www.maths.qmw.ac.uk /~bill/MAS427.html (648 words)

	Wedderburn's structure theorem
		For instance, it is not immediately clear why left semisimplicity is equivalent to right semisimplicity.
		Corollary 3.11 reduces the study of finite-dimensional semisimple algebras over a field to the study of finite-dimensional division algebras over this field.
		It is also possible to classify all finite-dimensional division rings over finite fields (they are all fields) but we are not going into this topic in this course.
	www.maths.warwick.ac.uk /~rumynin/rings2002/notes/node43.html (206 words)

	abstract3 (Site not responding. Last check: 2007-10-13)
		In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16.
		We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16.
		Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.
	web.syr.edu /~ykashina/abstract3.html (53 words)

	Scientific publications of Bernd Fiedler
		Fiedler, An Algorithm for the Decomposition of Ideals of Semi-Simple Rings and its Application to Symbolic Tensor Calculations by Computer, Universität Leipzig, Fakultät für Mathematik und Informatik, Leipzig, Germany, November 1999.
		Fiedler, An algorithm for the decomposition of ideals of the group ring of a symmetric group.
		Fiedler, Characterization of tensor symmetries by group ring subspaces and computation of normal forms of tensor coordinates.
	www.fiemath.de /publicat.htm (608 words)

	First year curriculum
		Rings, polynomial rings in one variable, unique factorization, non-commutative rings - matrix ring.
		Rings: commutative noetherian rings, Hilbert basis theorem, prime and maximal ideals and localizations, primary decomposition, integral extensions and normal rings, Dedekind domains, Eisenstein irreducibility criteria, group ring, semisimple rings and Wedderburn's theorem.
		Modules: tensor product, symmetric and exterior algebras and induced maps, exact functors, projective and injective modules, finitely generated modules over a Principal Ideal Domain with application to canonical forms of a matrix over a field, elementary theory of group representations.
	www.math.upenn.edu /grad/1stYearGrad.html (895 words)

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