On the set of catenary degrees of finitely generated cancellative commutative monoids

The catenary degree of an element [Formula: see text] of a cancellative commutative monoid [Formula: see text] is a nonnegative integer measuring the distance between the irreducible factorizations of [Formula: see text]. The catenary degree of the monoid [Formula: see text], defined as the supremum over all catenary degrees occurring in [Formula: see text], has been studied as an invariant of nonunique factorization. In this paper, we investigate the set [Formula: see text] of catenary degrees achieved by elements of [Formula: see text], focusing on the case where [Formula: see text] is finitely generated (where [Formula: see text] is known to be finite). Answering an open question posed by García-Sánchez, we provide a method to compute the smallest nonzero element of [Formula: see text] that parallels a well-known method of computing the maximum value. We also give several examples demonstrating certain extremal behavior for [Formula: see text].

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PRESENTATIONS OF FINITELY GENERATED SUBMONOIDS OF FINITELY GENERATED COMMUTATIVE MONOIDS

International Journal of Algebra and Computation ◽

10.1142/s021819670200105x ◽

2002 ◽

Vol 12 (05) ◽

pp. 659-670 ◽

Cited By ~ 3

Author(s):

J. C. ROSALES ◽

P. A. GARCÍA-SÁNCHEZ ◽

J. I. GARCÍA-GARCÍA

Keyword(s):

Commutative Monoid ◽

Finitely Generated ◽

Commutative Monoids ◽

Algorithmic Method ◽

Simple Equations ◽

Torsion Free

We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on ℕp and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple equations on a finitely generated commutative monoid.

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Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Discrete Applied Mathematics ◽

10.1016/j.dam.2006.03.013 ◽

2006 ◽

Vol 154 (14) ◽

pp. 1947-1959 ◽

Cited By ~ 8

Author(s):

S.T. Chapman ◽

P.A. García-Sánchez ◽

D. Llena ◽

J.C. Rosales

Keyword(s):

Linear Equations ◽

Finitely Generated ◽

Commutative Monoids ◽

Nonnegative Solutions ◽

Systems Of Linear Equations

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Locally graded groups with all subgroups normal-by-finite

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700037617 ◽

1996 ◽

Vol 60 (2) ◽

pp. 222-227 ◽

Cited By ~ 10

Author(s):

Howard Smith ◽

James Wiegold

Keyword(s):

Finite Group ◽

Finite Index ◽

Finitely Generated ◽

Locally Finite ◽

Residually Finite ◽

Residually Finite Group ◽

Open Question ◽

Locally Graded Groups

AbstractIn a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

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FINITELY VALUATIVE DOMAINS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501125 ◽

2012 ◽

Vol 11 (06) ◽

pp. 1250112 ◽

Cited By ~ 1

Author(s):

PAUL-JEAN CAHEN ◽

DAVID E. DOBBS ◽

THOMAS G. LUCAS

Keyword(s):

Nonnegative Integer ◽

Integral Domain ◽

Nonzero Element ◽

Integral Closure ◽

Quotient Field ◽

Prüfer Domain ◽

Maximal Ideals ◽

Saturated Chain

For a pair of rings S ⊆ T and a nonnegative integer n, an element t ∈ T\S is said to be within n steps of S if there is a saturated chain of rings S = S0 ⊊ S1 ⊊ ⋯ ⊊ Sm = S[t] with length m ≤ n. An integral domain R is said to be n-valuative (respectively, finitely valuative) if for each nonzero element u in its quotient field, at least one of u and u-1 is within n (respectively, finitely many) steps of R. The integral closure of a finitely valuative domain is a Prüfer domain. Moreover, an n-valuative domain has at most 2n + 1 maximal ideals; and an n-valuative domain with 2n + 1 maximal ideals must be a Prüfer domain.

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An Euler characteristic for modules of finite G-dimension

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15059 ◽

2008 ◽

Vol 102 (2) ◽

pp. 206 ◽

Cited By ~ 2

Author(s):

Sean Sather-Wagstaff ◽

Diana White

Keyword(s):

Euler Characteristic ◽

Betti Numbers ◽

Projective Dimension ◽

Finitely Generated ◽

Finite Projective Dimension ◽

Extremal Behavior ◽

Classical Invariant ◽

Category Of Modules ◽

Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

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On cominimaxness of generalized local cohomology modules

Tạp chí Khoa học ◽

10.54607/hcmue.js.16.3.2463(2019) ◽

2019 ◽

Vol 16 (3) ◽

pp. 50

Author(s):

Nguyen Thanh Nam ◽

Tran Tuan Nam ◽

Nguyen Minh Tri

Keyword(s):

Nonnegative Integer ◽

Local Cohomology ◽

Finitely Generated ◽

Local Cohomology Modules ◽

Generalized Local Cohomology Modules ◽

Cohomology Modules

This research introduces and focuses on (I, M)-cominimax modules. The paper shows that if t is an nonnegative integer, M is a finitely generated projective R-module and N is an R-module such that is minimax and is (I, M)-cominimax for all then is minimax and is finite.

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Irreducible decomposition of binomial ideals

Compositio Mathematica ◽

10.1112/s0010437x16007272 ◽

2016 ◽

Vol 152 (6) ◽

pp. 1319-1332 ◽

Cited By ~ 4

Author(s):

Thomas Kahle ◽

Ezra Miller ◽

Christopher O’Neill

Keyword(s):

Number Theory ◽

Duke Math ◽

Commutative Monoid ◽

Commutative Monoids ◽

Irreducible Decomposition ◽

Binomial Ideal ◽

Binomial Ideals ◽

Mesoprimary Decomposition

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

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PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

Journal of Algebra and Its Applications ◽

10.1142/s0219498803000593 ◽

2003 ◽

Vol 02 (04) ◽

pp. 435-449 ◽

Cited By ~ 17

Author(s):

ALBERTO FACCHINI ◽

FRANZ HALTER-KOCH

Keyword(s):

Commutative Ring ◽

Projective Modules ◽

Finitely Generated ◽

Commutative Monoids ◽

Isomorphism Classes

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

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