Rees Modules and Reduction of an Ideal Relative to a Noetherian Module

2011 ◽  
Vol 18 (spec01) ◽  
pp. 739-748
Author(s):  
Naser Zamani
Keyword(s):  
Local Ring ◽  
Proper Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Noetherian Module ◽  
Reduction Number ◽  
Necessary And Sufficient

Let (A, 𝔪) be a Noetherian local ring, E a non-zero finitely generated A-module, and 𝔟 a proper ideal of A. The aim of this paper is, under some assumptions on𝔟 and E, to clarify a necessary and sufficient condition for the Rees module of E associated to 𝔟 to obtain Cohen-Macaulayness. Also, among other independent results about the reduction number of 𝔟 relative to E, we expand a theorem of Marley.

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

scholarly journals On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

2015 ◽  
Vol 22 (spec01) ◽  
pp. 935-946 ◽  
Author(s):  
Majid Rahro Zargar ◽  
Hossein Zakeri
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Gorenstein Rings ◽  
Noetherian Local Ring ◽  
Dualizing Complex ◽  
Local Cohomology Modules ◽  
Gorenstein Injective ◽  
Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht M𝔞. We prove that injdim M < ∞ if and only if [Formula: see text] and that [Formula: see text]. We also prove that if R has a dualizing complex and Gid RM < ∞, then [Formula: see text]. Moreover if R and M are Cohen-Macaulay, then Gid RM < ∞ whenever [Formula: see text]. Next, for a finitely generated R-module M of dimension d, it is proved that if [Formula: see text] is Cohen-Macaulay and [Formula: see text], then [Formula: see text]. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.

Comaximal factorization of lifting ideals

2020 ◽  
pp. 2250044
Author(s):  
Esmaeil Rostami ◽  
Sina Hedayat ◽  
Reza Nekooei ◽  
Somayeh Karimzadeh
Keyword(s):  
Commutative Ring ◽  
Proper Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Approach ◽  
Ideal Formula ◽  
Basic Properties ◽  
Necessary And Sufficient

A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is called lifting whenever idempotents of [Formula: see text] lift to idempotents of [Formula: see text]. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal lifting ideals.

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

2015 ◽  
Vol 22 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Houyi Yu ◽  
Tongsuo Wu ◽  
Weiping Gu
Keyword(s):  
Local Ring ◽  
Complete Characterization ◽  
Local Rings ◽  
Star Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Star Graphs ◽  
Finite Local Ring ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

scholarly journals ON GENERATORS AND PRESENTATIONS OF SEMIDIRECT PRODUCTS IN INVERSE SEMIGROUPS

2009 ◽  
Vol 79 (3) ◽  
pp. 353-365 ◽  
Author(s):  
E. R. DOMBI ◽  
N. RUŠKUC
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Semidirect Products ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractIn this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

On Special Fiber Rings of Modules

2019 ◽  
Vol 72 (1) ◽  
pp. 225-242
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Residue Field ◽  
Independent Interest ◽  
General Situation ◽  
Finitely Generated ◽  
Fiber Ring ◽  
Noetherian Local Ring ◽  
Reduction Number

AbstractWe prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

scholarly journals On finitary properties for fiber products of free semigroups and free monoids

2020 ◽  
Vol 101 (2) ◽  
pp. 326-357
Author(s):  
Ashley Clayton
Keyword(s):  
Sufficient Conditions ◽  
Free Monoid ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Free Monoids ◽  
Fiber Product ◽  
Finite Presentability ◽  
Free Semigroups ◽  
Necessary And Sufficient

Abstract We consider necessary and sufficient conditions for finite generation and finite presentability for fiber products of free semigroups and free monoids. We give a necessary and sufficient condition on finite fiber quotients for a fiber product of two free monoids to be finitely generated, and show that all such fiber products are also finitely presented. By way of contrast, we show that fiber products of free semigroups over finite fiber quotients are never finitely generated. We then consider fiber products of free semigroups over infinite semigroups, and show that for such a fiber product to be finitely generated, the quotient must be infinite but finitely generated, idempotent-free, and $$\mathcal {J}$$ J -trivial. Finally, we construct automata accepting the indecomposable elements of the fiber product of two free monoids/semigroups over free monoid/semigroup fibers, and give a necessary and sufficient condition for such a product to be finitely generated.

Co-Cohen-Macaulay Modules and Generalized Local Cohomology

2011 ◽  
Vol 18 (spec01) ◽  
pp. 807-813
Author(s):  
Amir Mafi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Finite Dimension ◽  
Cohomological Dimension ◽  
Projective Dimension ◽  
Proper Ideal ◽  
Finitely Generated ◽  
Local Cohomology Module ◽  
Noetherian Local Ring ◽  
Finite Projective Dimension

Let (R,𝔪) be a Noetherian local ring, 𝔞 a proper ideal of R, and M, N two finitely generated R-modules of finite projective dimension m and of finite dimension n, respectively. It is shown that if n ≤ 2, then the generalized local cohomology module [Formula: see text] is a co-Cohen-Macaulay module. Additionally, we show that [Formula: see text] for all i > m+s and [Formula: see text], where s is the cohomological dimension of N with respect to 𝔞.

Restricted Burnside problem for semigroups and its application to language theory

2017 ◽  
Vol 10 (03) ◽  
pp. 1750045
Author(s):  
Y. Q. Guo ◽  
Shoufeng Wang ◽  
K. P. Shum
Keyword(s):  
Language Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Restricted Burnside Problem ◽  
Necessary And Sufficient ◽  
Torsion Property

In this paper, we introduce the concept of strongly torsion property of semigroups. Then we prove that the strongly torsion property of a semigroup is a necessary and sufficient condition for a finitely generated semigroup to be finite, and hence, the strongly torsion property is equivalent to the torsion property together with the permutation property for the class of finitely generated semigroups. Finally, we give an application of our main result to language theory. A question proposed in [H. Prodinger, Congruences defined by languages and filters, Inf. Control 44 (1980) 36–46] is consequently answered.

scholarly journals SOME REMARKS ON TOPOLOGICAL FULL GROUPS OF CANTOR MINIMAL SYSTEMS

2006 ◽  
Vol 17 (02) ◽  
pp. 231-251 ◽  
Author(s):  
HIROKI MATUI
Keyword(s):  
Sufficient Condition ◽  
Full Group ◽  
Normal Subgroups ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
Cantor Minimal Systems

Giordano, Putnam and Skau showed that topological full groups of Cantor minimal systems are complete invariants for flip conjugacy. We will completely determine the structure of normal subgroups of the topological full group. Moreover, a necessary and sufficient condition for the topological full group to be finitely generated will be given.

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