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Topic: Dirichlet ring

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Peter Gustav Lejeune Dirichlet

1837 in science

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1920

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	Unit (ring theory) - Wikipedia, the free encyclopedia
		In mathematics, a unit in a (unital) ring R is an invertible element of u, i.e.
		In a commutative unital ring R, the group of units U(R) acts on R via multiplication.
		For example, in the ring Z of integers, n and −n are associates.
	en.wikipedia.org /wiki/Unit_(ring_theory) (323 words)

	Dirichlet convolution - Wikipedia, the free encyclopedia
		In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is of importance in number theory.
		With addition and Dirichlet convolution, the set of arithmetic functions forms a commutative ring with multiplicative identity ε, the Dirichlet ring (note that it is not a field because some arithmetic functions do not have Dirichlet inverses).
		The units of this ring are the arithmetical functions f with f(1) ≠ 0.
	en.wikipedia.org /wiki/Dirichlet_convolution (385 words)

	Ring Theory
		In this article we shall be concerned with the development of the theory of commutative rings (that is rings in which multiplication is commutative) and the theory of non-commutative rings up to the 1940's.
		It is important to realise that at this stage rings of polynomials and rings of numbers were being studied, but it was to be another 40 years before an axiomatic theory of commutative rings was to be developed bringing these theories together.
		In contrast to commutative ring theory, which as we have seen grew from number theory, non-commutative ring theory developed from an idea which, at the time of its discovery, was heralded as a great advance in applied mathematics.
	www-groups.dcs.st-and.ac.uk /~history/PrintHT/Ring_theory.html (1857 words)

	Ring Theory
		However, axioms for rings are not given by Weber and the axiomatic treatment of commutative rings was not developed until the 1920's in the work of Emmy Noether and Krull.
		The greatest early contributor to the theory of non-commutative rings was the Scottish mathematician Wedderburn.
		In 1908 Wedderburn had the important idea of splitting the study of a ring into two parts, one part he called the radical, the part which was left being called semi-simple.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html (1857 words)

	Dedekind (Site not responding. Last check: 2007-11-05)
		Gauss died in 1855 and Dirichlet was appointed to fill the vacant chair at Göttingen.
		Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
		Dedekind's study of Dirichlet's work did, in fact, lead to his own study of algebraic number fields, as well as to his introduction of ideals.
	www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Dedekind.html (1962 words)

	PlanetMath: algebraic number theory
		It is a well-known fact that the ring of integers of a number field is a Dedekind domain.
		The structure of the unit group is described by Dirichlet's unit theorem, which asserts the existence of a system of fundamental units.
		An application of Dirichlet's unit theorem: units of quadratic fields.
	planetmath.org /encyclopedia/AlgebraicNumberTheory.html (936 words)

	Creation Functions
		The group of Dirichlet characters mod N with image in the order-r cyclic subgroup of the ring R generated by the root of unity z.
		This is a Dirichlet character of modulus equal to the least common multiple of the moduli of x and y.
		The base rings and chosen roots of unity of the parents of x and y are equal.
	www.math.lsu.edu /magma/text1262.htm (1976 words)

	Volume 49, number 1 (2005) (Site not responding. Last check: 2007-11-05)
		We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one.
		In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series.
		Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.
	mat.uab.es /~pubmat/v49(1)/49105_04.html (76 words)

	Master's and senior theses advised by Tim Hsu (Site not responding. Last check: 2007-11-05)
		In this (original) thesis, the taping number is defined rigorously, and the taping numbers of all of the regular polyhedra are determined, using a combination of combinatorics, geometry, and calculus on manifolds.
		The Dirichlet problem on a network of resistors is: Given a network of known resistors and batteries, determine the voltages at all nodes (or equivalently, the currents on all resistors).
		The inverse Dirichlet problem is: Given a resistor network of known topology but unknown resistance, figure out the resistances by attaching appropriate batteries and measuring the resulting currents.
	www.mathcs.sjsu.edu /faculty/Moved/hsu/thesislist.html (773 words)

	Dirichlet, Peter Gustav Lejeune -- Encyclopædia Britannica
		He taught at the universities of Breslau (1827) and Berlin (1828–55) and in 1855 succeeded Carl Friedrich Gauss at the University of Göttingen.
		More results on "Dirichlet, Peter Gustav Lejeune" when you join.
		German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into a product of primes) to complex number fields.
	www.britannica.com /eb/article-9030603?tocId=9030603 (735 words)

	Kummer (Site not responding. Last check: 2007-11-05)
		This led to Jacobi, and later Dirichlet, corresponding with Kummer on mathematical topics and they soon realised the great potential for the highest level of mathematics that Kummer possessed.
		In 1840 Kummer married a cousin of Dirichlet's wife.
		In 1842, with strong support from Jacobi and Dirichlet, he was appointed a full professor at the University of Breslau, now Wroclaw in Poland.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Kummer.html (1372 words)

	Silberberg (Site not responding. Last check: 2007-11-05)
		The paper studies the structure of the ring A of arithmetical functions, where the multiplication is defined as the Dirichlet convolution.
		It is proven that A itself is not a discrete valuation ring, but a certain extension of it is constructed,this extension being a discrete valuation ring.
		Finally, the metric structure of the ring A is examined.
	www.mat.ub.es /EMIS/journals/AM/00-2/silber.htm (99 words)

	Introduction
		A Dirichlet character is a homomorphism varepsilon:(Z/NZ)^ * -> C^ * of abelian groups.
		Dirichlet characters are of interest because they decompose M_k(Gamma_1(N)) into more manageable chunks.
		For any ring R, we define the space of modular forms over R to be M_R = M_Z tensor_(Z) R. Thus M_R is a free R-module of rank d with basis the images of f_1,..., f_d in M_Z tensor_(Z) R. The computation of M_R is discussed in Section Base Extension.
	www.math.lsu.edu /magma/text1301.htm (1519 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.
		Also, A is said to be an EGD ring (resp., EUGD ring) if the inclusion map from A to B satisfies GD (resp., is universally going-down) for each overring B of A. The concept of going-down ring (resp., universall going-down ring) is not equivalent to the concept of EGD ring (resp., EUGD ring).
	www.math.uh.edu /~hjm/Vol29-3.html (2232 words)

	Bodkin Associates (Site not responding. Last check: 2007-11-05)
		For example in the ring Z of integers n and− n are Associtaes.
		In fact that is the source for the unit terminology — which shouldn't be confused with the 'unit' of unital rings.
		One can check that U is a functor from the category of rings, to the category of groups: a ringhomomorphism must map units to units.
	www.super8filmmaking.com /tail/34817-bodkin-associates.html (184 words)

	Prime number (Site not responding. Last check: 2007-11-05)
		In the context of ring theory, a branch of abstract algebra, the term "prime element" has a specific meaning, and under this meaning, the additive inverse of any prime number is also prime.
		For the ring of integers, the set of prime elements equals the set of irreducible elements; it's {...−11, −7, −5, −3, −2, 2, 3, 5, 7, 11,...}.
		The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11),...
	www.free-download-soft.com /info/prime-numbers.html (2261 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Introduction Let l be an odd prime number and O a ring of integers that contains the lth roots of unity, has only one prime ideal above l and its Picard group has no l-torsion.
		Let A be the ring O[1_l] regarded as a fixed subring of the field C of complex numbers and consider the etale ap- On proximation map BGLn(A) -!
		Also, we denote H(-; Fl) by H(-) whenever it is not otherwise stated, and we keep the notation 2r2 for the number of complex embeddings of A. A Useful Lemma The goal of this section is to prove lemma 4 that will be needed later.
	hopf.math.purdue.edu /Anton/etaleappr.txt (1877 words)

	PlanetMath: examples of groups
		The units of any ring form a group with respect to the ring multiplication; e.g.
		Generalizing the last two examples, every ring (and every monoid) contains a group, its group of units (invertible elements), where the group operation is ring (monoid) multiplication.
		The set of arithmetic functions that take a value other than 0 at 1 form an Abelian group under Dirichlet convolution.
	planetmath.org /encyclopedia/ExamplesOfGroups.html (905 words)

	Program Files\Netscape\Communicator\Program\dedexxx
		In 1855 Gauss died and Dirichlet was appointed to fill the vacant chair at Gottingen.
		While working with Dirichlet Dedekind soon became close friends, while also rubbing some of his mathematical interests onto Dedekind.
		This led to the unique factorisation of integers into prime powers to be generalized to ideals in other rings.
	www.andrews.edu /~calkins/math/biograph/biodedek.htm (1253 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The formula is deduced from an expansion of the Eisenstein series which is closely related to the Fourier-Jacobi expansion and which has the form of an infinite sum of terms indexex by primitive adelic theta function.
		Each term is a product of (1) an explicit monomial of Hecke and Dirichlet L-series for the specified character and weight, and (2) a theta function obtained by applying formal Dirichlet series of operators to the indexed adelic theta function.
		The Dirichlet series consists of operators on the graded ring of adelic theta functions and can be expressed in terms of the eigenvalues of the Eisenstein series by a result of Shintani.
	www.cs.brandeis.edu /~tim/Papers/Dissertation/abstract (255 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Let F be the vector space of formal linear combinations sum_m a_m D_m of elements in M with real or complex coefficients.
		Then F becomes a ring under termwise addition and multiplication defined by bilinearity using the multiplicative rule (4): (sum_m a_m D_m)(sum_n b_n D_n) := sum_{m,n} a_m b_n D_{mn} (5) = sum_n (sum_{dn} a_d b_{n/d}) D_n The units in the ring are just the b = sum_n b_n D_n with b_1
		This ring can also be viewed as the space S of sequences a = (a_1, a_2,....) under termwise addition and the "convolution product" (a*b)_n := sum_{dn} a_d b_{n/d} with unit (1, 0, 0,...).
	www.math.niu.edu /~rusin/papers/known-math/99/convolution_disc (884 words)

	Tornado Siren Location, Ann Arbor, Michigan (Site not responding. Last check: 2007-11-05)
		The red outlines of polygons, in a sort of bubble foam, are outlines of the Dirichlet tesselation on the fire stations.
		The Dirichlet polygons are mutually exclusive and cover the entire area in the one mile buffer.
		Outside the Dirichlet tesselation, highest priority might therefore be given to the gap at the right edge of the tesselation that is within the freeway ring but is as yet uncovered by a siren.
	www-personal.umich.edu /~sarhaus/image/solstice/sum03/sandy/Tornado_Siren_Location,_Ann_Arbor,_Michigan.html (798 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In this course, all rings are commutative.'' (I never needed to explicitly refer to a noncommutative ring.
		With those conventions: A ring is any object with 0 1 + - * that satisfies every identity satisfied by the integers.
		This fact should be stated in algebra courses, and some students figure it out on their own, but it can't hurt to say it again.
	cr.yp.to /1995-514/thoughts.html (361 words)

	Dirichlet Characters
		The Dirichlet characters package provides support for computing with the group of homomorphism
		Dirichlet group over a field K as the set of rational characters from
		Dirichlet characters have been developed in support of the new Hecke module types.
	magma.maths.usyd.edu.au /magma/Features/node232.html (72 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Prerequisites: A basic grounding in algebraic concepts like groups, rings, fields, basic Galois theory, and a basic course in analysis (like 3210).
		Studying rational points for instance leads to very delicate questions about number fields: for instance whether the ring ${\bf Z}[\zeta_p]$ is a unique factorisation domain, where $\zeta_p$ is a $p$th root of unity.
		The textbook used for the course provides a hands-on experience of studying the basics of algebraic number theory via many examples.
	www.math.utah.edu /~shekhar/6350.html (137 words)

	MTH 617 Algebraic number theory, Summer semester 2002 (Site not responding. Last check: 2007-11-05)
		I would like to introduce some simple (?) properties of number fields (finite (and hence algebraic) extensions of the field of rational numbers) and their rings of integers.
		Knowledge of algebra and linear algebra (groups, rings, fields and vector spaces) is required.
		Characters of finite Abelian groups, Dirichlet series, Dirichlet's theorem on primes in arithmetic progression.
	home.iitk.ac.in /~abhijit/course/MTH617/SS02 (719 words)

	Evidence for a Spectral Interpretation of the Zeros of L-Functions - Rubinstein (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		...functions with real Dirichlet characters described at (22) the low lying zeros appear to show symplectic symmetry.
		1 The evaluation of Dirichlet L-functions (context) - Davies, Haselgrove - 1961
		1 Dirichlet series in the critical strip (context) - Keiper, zeros et al.
	citeseer.ist.psu.edu /91989.html (810 words)

	Dept. of Mathematics, IIT Kanpur (Site not responding. Last check: 2007-11-05)
		Commutative rings; Ideals, prime and maximal ideals, Noetherian Artinian rings, Primary decomposition in Noetherian rings, Modules over commutative rings; Exact sequences, the Hom and Tensor functors, rings and modules of fractions, Integral dependence, valuations and Dedekind domains.
		Polynomial rings over fields, Extension of fields, computation in GF(q), Root fields of polynomials, Vector space over finite fields, Binary group codes, Hamming codes, polynomial codes, Linear block codes, The structure of cyclic codes, Quadratic residue codes, Reed Mueller codes, Simplex codes.
		Normal families and applications, Riemann mapping theorem, Conformal mapping of a sequence of domains; Modular function, Hyperbolic metric, Elementary theory of univalent functions, Lowner's theory, Dirichlet problem, Green's function and conformal mapping; transfinite diameter and capacity; Symmetrization, Extremal length and prime ends.
	www.iitk.ac.in /math/601_650.htm (990 words)

	Mathematics Magazine: April 2005
		The functions from the positive integers to the real numbers or complex numbers form a ring under pointwise addition and the Dirichlet product.
		Ideas from Abelian group theory are applied to the group of units of this ring, with interesting results.
		It is shown that the functions withf(1) = 1 form a vector space over the rationals, and that the multiplicative functions form a subspace.
	www.maa.org /pubs/mag_apr05_toc.html (867 words)

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