scholarly journals Divergence type of some subgroups of finitely generated Fuchsian groups

1981 ◽  
Vol 1 (2) ◽  
pp. 209-221 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Critical Exponent ◽  
Hyperbolic Plane ◽  
Sufficient Conditions ◽  
Discrete Subgroup ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Parabolic Element ◽  
Divergence Type ◽  
Necessary And Sufficient

AbstractLet Г be a finitely generated discrete subgroup of the isometries of the hyperbolic plane H2 with at least one parabolic element. We prove that, if Г1 is a subgroup of Г with Г/Г1 abelian, the ‘critical exponent’ of Г1 is the same as that of Г. We give necessary and sufficient conditions-in terms of the rank of Г/Г1, the critical exponent of Г, and the image of parabolic elements of Г in Г/Г1 - for the Poincaré series of Г1 to diverge at the critical exponent.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

scholarly journals On the equivalence of cancellative extensions of commutative cancellative semigroups by groups

1971 ◽  
Vol 12 (2) ◽  
pp. 187-192
Author(s):  
Charles V. Heuer
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ideal Extension ◽  
Cancellative Semigroup ◽  
Finitely Generated ◽  
Necessary And Sufficient

In [1] D. W. Miller and the author established necessary and sufficient conditions for the existence of a cancellative (ideal) extension of a commutative cancellative semigroup by a cyclic group with zero. The purpose of this paper is to extend these results to cancellative extensions by any finitely generated Abelian group with zero and to establish in this general case conditions under which two such extensions are equivalent.

scholarly journals Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Khuanchanok Chaichana ◽  
Pradthana Jaipong
Keyword(s):  
Prime Number ◽  
Fuchsian Group ◽  
Continued Fraction ◽  
Sufficient Conditions ◽  
Minimal Length ◽  
Necessary And Sufficient Conditions ◽  
Fuchsian Groups ◽  
Suborbital Graphs ◽  
Necessary And Sufficient ◽  
Recursive Representation

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

1988 ◽  
Vol 31 (3) ◽  
pp. 374-379 ◽  
Author(s):  
Kenneth G. Wolfson
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Free Module ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Ore Domain ◽  
Necessary And Sufficient

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

EXCHANGE RINGS OVER WHICH EVERY REGULAR MATRIX ADMITS DIAGONAL REDUCTION

2004 ◽  
Vol 03 (02) ◽  
pp. 207-217 ◽  
Author(s):  
HUANYIN CHEN
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
Regular Matrix ◽  
Finitely Generated ◽  
Exchange Rings ◽  
Necessary And Sufficient ◽  
Diagonal Reduction

In this paper, we investigate the necessary and sufficient conditions on exchange rings, under which every regular matrix admits diagonal reduction. Also we show that an exchange ring R is strongly separative if and only if for any finitely generated projective right R-module C, if A and B are any right R-modules such that 2C⊕A≅C⊕B, then C⊕A≅B.

scholarly journals A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane ℍ 2 into ℝ 4

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Mor Dione ◽  
Bira Gaye Diop ◽  
Athoumane Niang
Keyword(s):  
Isometric Immersion ◽  
Hyperbolic Plane ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Interpretation ◽  
Open Set ◽  
Local Solutions ◽  
Necessary And Sufficient ◽  
Complete Solutions

A simple interpretation of the PDE for isometric immersion of the hyperbolic plane ℍ 2 into ℝ 4 is given. Thus, necessary and sufficient conditions are obtained. And, under natural additional conditions, we show that there are no complete solutions, but we can have special local solutions.

FREE TERNARY ALGEBRAS

2000 ◽  
Vol 10 (06) ◽  
pp. 739-749 ◽  
Author(s):  
RAYMOND BALBES
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Free Generator ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Bounded Distributive Lattice ◽  
Ternary Algebra ◽  
Free Generators ◽  
Ternary Algebras ◽  
Necessary And Sufficient

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

Projectivity and flatness over the endomorphism ring of a finitely generated quasi-Poisson comodule

2014 ◽  
Vol 13 (08) ◽  
pp. 1450060
Author(s):  
T. Guédénon
Keyword(s):  
Vector Space ◽  
Endomorphism Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Module Algebra ◽  
Finitely Generated ◽  
Enveloping Algebra ◽  
Grouplike Element ◽  
Necessary And Sufficient ◽  
Dual Ring

Let k be a field of characteristic 0, A a noncommutative Poisson k-algebra, U(A) the ordinary enveloping algebra of A, 𝒞 a quasi-Poisson A-coring that is projective as a left A-module, *𝒞 the left dual ring of 𝒞 (it is a right U(A)-module algebra) and Λ a right quasi-Poisson 𝒞-comodule that is finitely generated as a right U(A)#*𝒞-module. The vector space End 𝒫,𝒞(Λ) of right quasi-Poisson 𝒞-colinear maps from Λ to Λ is a ring. We give necessary and sufficient conditions for projectivity and flatness of a module over End 𝒫,𝒞(Λ). If 𝒞 contains a fixed quasi-Poisson grouplike element, we can replace Λ with A.

An alternative perspective on flatness of modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650145 ◽  
Author(s):  
Yılmaz Durğun
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Injective Modules ◽  
Alternative Perspective ◽  
Necessary And Sufficient

Given modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be absolutely [Formula: see text]-pure if [Formula: see text] is a monomorphism for every extension [Formula: see text] of [Formula: see text]. For a module [Formula: see text], the absolutely pure domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is absolutely [Formula: see text]-pure. As an opposite to flatness, a module [Formula: see text] is said to be f-indigent if its absolutely pure domain is smallest possible, namely, consisting of exactly the fp-injective modules. Properties of absolutely pure domains and off-indigent modules are studied. In particular, the existence of f-indigent modules is determined for an arbitrary rings. For various classes of modules (such as finitely generated, simple, singular), necessary and sufficient conditions for the existence of f-indigent modules of those types are studied. Furthermore, f-indigent modules on commutative Noetherian hereditary rings are characterized.

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