Modules and abelian groups with a bounded domain of injectivity

2018 ◽  
Vol 17 (06) ◽  
pp. 1850108 ◽  
Author(s):  
Yilmaz Mehmet Demirci
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Direct Summand ◽  
Commutative Ring ◽  
Bounded Domain ◽  
Abelian Groups ◽  
Prime Integer

In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group [Formula: see text] is impecunious if and only if for every prime integer [Formula: see text], [Formula: see text] has a direct summand isomorphic to [Formula: see text] for some positive integer [Formula: see text]. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

Spectral synthesis and residually finite-dimensional algebras

2017 ◽  
Vol 16 (10) ◽  
pp. 1750200 ◽  
Author(s):  
László Székelyhidi ◽  
Bettina Wilkens
Keyword(s):  
Spectral Analysis ◽  
Abelian Group ◽  
Theoretical Approach ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Analysis And Synthesis ◽  
Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

scholarly journals Pointed finite tensor categories over abelian groups

2017 ◽  
Vol 28 (11) ◽  
pp. 1750087
Author(s):  
Iván Angiono ◽  
César Galindo
Keyword(s):  
Abelian Group ◽  
Cohomology Class ◽  
Hopf Algebras ◽  
Abelian Groups ◽  
Tensor Category ◽  
Tensor Categories ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

Hypercyclic Abelian Groups of Affine Maps on ℂn

2013 ◽  
Vol 56 (3) ◽  
pp. 477-490 ◽  
Author(s):  
Adlene Ayadi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Dense Orbit ◽  
Finitely Generated ◽  
Affine Maps

Abstract.We give a characterization of hypercyclic abelian group 𝒢 of affine maps on ℂn. If G is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by n affine maps on Cn has a dense orbit.

scholarly journals Rainbow-free $3$-colorings of Abelian Groups

10.37236/2054 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Amanda Montejano ◽  
Oriol Serra
Keyword(s):  
Abelian Group ◽  
Arithmetic Progression ◽  
Structural Characterization ◽  
Abelian Groups ◽  
Cyclic Groups ◽  
Chromatic Class

A $3$-coloring of the elements of an abelian group is said to be rainbow-free if there is no $3$-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow-free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free $3$-coloring of cyclic groups.

Spectra of independent power measures

1972 ◽  
Vol 72 (1) ◽  
pp. 27-35 ◽  
Author(s):  
W. J. Bailey ◽  
G. Brown ◽  
W. Moran
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Elementary Proof ◽  
Natural Generalization ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Power Measure ◽  
Lca Groups

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

Automorphisms of a Certain Skew Polynomial Ring of Derivation Type

1990 ◽  
Vol 42 (6) ◽  
pp. 949-958
Author(s):  
Isao Kikumasa
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Prime Integer ◽  
Ring Homomorphisms ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Throughout this paper, all rings have the identity 1 and ring homomorphisms are assumed to preserve 1. We use p to denote a prime integer and F to denote a field of characteristic p. For an element α in F, we setA = F[ϰ]/(ϰp - α)F[ϰ].Moreover, by D and R, we denote the derivation of A induced by the ordinary derivation of F[ϰ] and the skew polynomial ring A[X,D] where aX = Xa+D(a) (a ∈ A), respectively (cf. [2]).In [3], R. W. Gilmer determined all the B-automorphisms of B[X] for any commutative ring B. Since then, some extensions or generalizations of his results have been obtained ([1], [2] and [5]). As to the characterization of automorphisms of skew polynomial rings, M. Rimmer [5] established a thorough result in case of automorphism type, while M. Ferrero and K. Kishimoto [2], among others, have made some progress in case of derivation type.

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