A characterization of duo-rings in which every Dedekind finite module is finitely generated

2014 ◽  
Vol 9 ◽  
pp. 733-736 ◽  
Author(s):  
Sidy Demba Toure ◽  
Khady Diop ◽  
Sidy Mohamed Ould Mohamed ◽  
Mamadou Sanghare
Keyword(s):  
Finitely Generated ◽  
Finite Module

scholarly journals Residual dimension of nilpotent groups

2020 ◽  
Vol 23 (5) ◽  
pp. 801-829
Author(s):  
Mark Pengitore
Keyword(s):  
New York ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Nontrivial Element ◽  
Maximum Order ◽  
Asymptotic Bounds ◽  
Group Morphism ◽  
A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

2003 ◽  
Vol 31 (9) ◽  
pp. 4195-4214 ◽  
Author(s):  
Alberto Facchini ◽  
Dolors Herbera ◽  
Iskhak Sakhajev
Keyword(s):  
Finitely Generated ◽  
Flat Modules

Abelian Semigroups of Matrices on ℂn and Hypercyclicity

2013 ◽  
Vol 57 (2) ◽  
pp. 323-338 ◽  
Author(s):  
Adlene Ayadi ◽  
Habib Marzougui
Keyword(s):  
Normal Form ◽  
Complete Characterization ◽  
Finitely Generated ◽  
Abelian Semigroup

AbstractWe give a complete characterization of a hypercyclic abelian semigroup of matrices on ℂn. For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over ℂ that form a hypercyclic abelian semigroup on ℂn. In particular, we show that no abelian semigroup generated by n matrices on ℂn can be hypercyclic.

scholarly journals A Characterization of Gorenstein Rings and Grothendieck's Conjecture

2009 ◽  
Vol 16 (04) ◽  
pp. 653-660
Author(s):  
Kazem Khashyarmanesh
Keyword(s):  
Local Ring ◽  
Regular Sequence ◽  
Negative Integer ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Filter Regular Sequence ◽  
System Of Parameters

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

scholarly journals Bounded endomorphisms of free P-algebras

1992 ◽  
Vol 34 (2) ◽  
pp. 209-214
Author(s):  
Daniel Ševčovič
Keyword(s):  
Free Algebra ◽  
Present Note ◽  
Finitely Generated ◽  
Efficient Tool ◽  
Pseudocomplemented Lattices

The present note deals with bounded endomorphisms of free p-algebras (pseudocomplemented lattices). The idea of bounded homomorphisms was introduced by R. McKenzie in [8]. T. Katriňák [5] subsequently studied the properties of bounded homomorphisms for the varieties of p-algebras. This concept is also an efficient tool for the characterization of, so-called, splitting as well as projective algebras in the varieties of all lattices or p-algebras. For details the reader is referred to [2], [5], [6], [7] and other references therein. Let us emphasize that the main results that are contained in the above mentioned references strongly depend on the boundedness of each endomorphism of any finitely generated free algebra in a given variety.

scholarly journals Injectives in finitely generated universal Horn classes

1987 ◽  
Vol 52 (3) ◽  
pp. 786-792
Author(s):  
Michael H. Albert ◽  
Ross Willard
Keyword(s):  
Finitely Generated ◽  
Finite Structures ◽  
Finite Set

AbstractLet K be a finite set of finite structures. We give a syntactic characterization of the property: every element of K is injective in ISP(K). We use this result to establish that is injective in ISP() for every two-element algebra .

Rings with A Finitely Generated Total Quotient Ring

1974 ◽  
Vol 17 (1) ◽  
pp. 1-4 ◽  
Author(s):  
John Conway Adams
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Rank One ◽  
Definition Of

Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.

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