MV-modules of fractions

2019 ◽  
Vol 19 (07) ◽  
pp. 2050131
Author(s):  
Hoda Banivaheb ◽  
Arsham Borumand Saeid
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Ideal Formula ◽  
One To One

In this paper, we study the notion of [Formula: see text]-modules of fractions with respect to a ⋅-prime ideal [Formula: see text]. Also the relation between the ideals of [Formula: see text]-module [Formula: see text] and the ideals of [Formula: see text]-module of fractions [Formula: see text] is studied and it is shown that there is a one to one correspondence between [Formula: see text]-prime [Formula: see text]-ideals of [Formula: see text]-module [Formula: see text] and [Formula: see text]-prime [Formula: see text]-ideals of [Formula: see text]-module of fractions [Formula: see text], where [Formula: see text] is the maximal localization of [Formula: see text] at a maximal ⋅-ideal [Formula: see text] of [Formula: see text] and [Formula: see text] is a ⋅-prime ideal of [Formula: see text] such that [Formula: see text].

scholarly journals On 2-absorbing ideals of commutative semirings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050034
Author(s):  
H. Behzadipour ◽  
P. Nasehpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Minimal Prime ◽  
Unique Maximal Ideal

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if [Formula: see text] is a nonzero proper ideal of a subtractive valuation semiring [Formula: see text] then [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if [Formula: see text] or [Formula: see text] where [Formula: see text] is a prime ideal of [Formula: see text]. We also show that each 2-absorbing ideal of a subtractive semiring [Formula: see text] is prime if and only if the prime ideals of [Formula: see text] are comparable and if [Formula: see text] is a minimal prime over a 2-absorbing ideal [Formula: see text], then [Formula: see text], where [Formula: see text] is the unique maximal ideal of [Formula: see text].

scholarly journals NONCOMMUTATIVE HILBERT RINGS

2004 ◽  
Vol 03 (04) ◽  
pp. 437-443 ◽  
Author(s):  
ALGIRDAS KAUCIKAS ◽  
ROBERT WISBAUER
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
The Other ◽  
Prime Ideals ◽  
Natural Form ◽  
Noncommutative Rings ◽  
Ideal Formula ◽  
Contraction Property ◽  
Maximal Ideals

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Noetherian domains in which every ideal is isomorphic to a trace ideal

2019 ◽  
Vol 19 (10) ◽  
pp. 2050200
Author(s):  
A. Mimouni
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Large Family ◽  
One Dimensional ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Trace Ideal

This paper seeks an answer to the following question: Let [Formula: see text] be a Noetherian ring with [Formula: see text]. When is every ideal isomorphic to a trace ideal? We prove that for a local Noetherian domain [Formula: see text] with [Formula: see text], every ideal is isomorphic to a trace ideal if and only if either [Formula: see text] is a DVR or [Formula: see text] is one-dimensional divisorial domain, [Formula: see text] is a principal ideal of [Formula: see text] and [Formula: see text] posses the property that every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. Next, we globalize our result by showing that a Noetherian domain [Formula: see text] with [Formula: see text] has every ideal isomorphic to a trace ideal if and only if either [Formula: see text] is a PID or [Formula: see text] is one-dimensional divisorial domain, every invertible ideal of [Formula: see text] is principal and for every non-invertible maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] is a principal ideal of [Formula: see text] and every ideal of [Formula: see text] is isomorphic to a trace ideal of [Formula: see text]. We close the paper by examining some classes of non-Noetherian domains with this property to provide a large family of original examples.

scholarly journals Fuzzy Normed Rings

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 515 ◽  
Author(s):  
Aykut Emniyet ◽  
Memet Şahin
Keyword(s):  
Prime Ideal ◽  
Fuzzy Sets ◽  
Maximal Ideal ◽  
Ring Homomorphism ◽  
Ring Theory ◽  
Basic Properties ◽  
Normed Ring ◽  
Normed Ideal ◽  
Algebraic Properties ◽  
Definition Of

In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.

On the number of incongruent solutions to a quadratic congruence over algebraic integers

2019 ◽  
Vol 15 (01) ◽  
pp. 105-130
Author(s):  
Ramy F. Taki Eldin
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Exponential Sums ◽  
Nonzero Ideal ◽  
Explicit Formulas ◽  
Algebraic Integers ◽  
Ideal Formula ◽  
Quadratic Congruence ◽  
Ring Of Algebraic Integers

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.

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