Families of generalized Cohen–Macaulay and filter rings

2021 ◽  
pp. 2250214
Author(s):  
Y. Azimi ◽  
N. Shirmohammadi
Keyword(s):  
Commutative Ring ◽  
Rees Algebra ◽  
The Family

Let [Formula: see text] be a commutative ring with unity, [Formula: see text] and [Formula: see text] an ideal of [Formula: see text]. Define [Formula: see text] to be [Formula: see text] a quotient of the Rees algebra. In this paper, we investigate when the rings in the family are generalized Cohen–Macaulay or filter rings and show that these properties are independent of the choice of [Formula: see text] and [Formula: see text].

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

On Operators and Distributions

1968 ◽  
Vol 11 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Raimond A. Struble
Keyword(s):  
Commutative Ring ◽  
Operational Calculus ◽  
Delta Function ◽  
Generalized Functions ◽  
Dirac Delta Function ◽  
Quotient Field ◽  
Dirac Delta ◽  
The Family ◽  
Continuous Complex ◽  
Complex Valued

Mikusinski [1] has extended the operational calculus by methods which are essentially algebraic. He considers the family C of continuous complex valued functions on the half-line [0,∞). Under addition and convolution C becomes a commutative ring. Titchmarsh's theorem [2] shows that the ring has no divisors of zero and, hence, that it may be imbedded in its quotient field Q whose elements are then called operators. Included in the field are the integral, differential and translational operators of analysis as well as certain generalized functions, such as the Dirac delta function. An alternate approach [3] yields a rather interesting result which we shall now describe briefly.

scholarly journals New algebraic properties of quadratic quotients of the Rees algebra

2019 ◽  
Vol 18 (03) ◽  
pp. 1950047 ◽  
Author(s):  
Marco D’Anna ◽  
Francesco Strazzanti
Keyword(s):  
Prime Ideal ◽  
Integral Domain ◽  
Rees Algebra ◽  
Ideal Formula ◽  
Local Case ◽  
The Family ◽  
Algebraic Properties

We study some properties of a family of rings [Formula: see text] that are obtained as quotients of the Rees algebra associated with a ring [Formula: see text] and an ideal [Formula: see text]. In particular, we give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen–Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when [Formula: see text] is an integral domain, reduced, quasi-Gorenstein, or satisfies Serre’s conditions.

Some algebraic and homological properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (12) ◽  
pp. 1950232
Author(s):  
Mahnaz Salek ◽  
Elham Tavasoli ◽  
Abolfazl Tehranian ◽  
Maryam Salimi
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Ideal Formula ◽  
Von Neumann Regular ◽  
Semisimple Ring ◽  
The Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a proper ideal of [Formula: see text]. In this paper, we study some algebraic and homological properties of a family of rings [Formula: see text], with [Formula: see text], that are obtained as quotients of the Rees algebra associated with the ring [Formula: see text] and the ideal [Formula: see text]. Specially, we study when [Formula: see text] is a von Neumann regular ring, a semisimple ring and a Gaussian ring. Also, we study the classical global and weak global dimensions of [Formula: see text]. Finally, we investigate some homological properties of [Formula: see text]-modules and we show that [Formula: see text] and [Formula: see text] are Gorenstein projective [Formula: see text]-modules, provided some special conditions.

scholarly journals Biquandle virtual brackets

2019 ◽  
Vol 28 (11) ◽  
pp. 1940003 ◽  
Author(s):  
Sam Nelson ◽  
Kanako Oshiro ◽  
Ayaka Shimizu ◽  
Yoshiro Yaguchi
Keyword(s):  
Commutative Ring ◽  
Knot Theory ◽  
Infinite Family ◽  
Polynomial Form ◽  
Link Diagrams ◽  
Classical Definition ◽  
Special Cases ◽  
The Family

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

Quasi-primaryful modules

2015 ◽  
Vol 08 (03) ◽  
pp. 1550051
Author(s):  
Hosein Fazaeli Moghimi ◽  
Mahdi Samiei
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Artinian Ring ◽  
The Other ◽  
Noetherian Rings ◽  
Projective Modules ◽  
New Class ◽  
The Family ◽  
Free Modules ◽  
Finitely Generated Modules

Let [Formula: see text] be a commutative ring with identity. The purpose of this paper is to introduce and to study a new class of modules over [Formula: see text] called quasi-primaryful [Formula: see text]-modules. This class contains the family of finitely generated modules properly, on the other hand it is contained in the family of primeful [Formula: see text]-modules properly, and three concepts coincide if they are multiplication modules. We show that free modules, projective modules over domains and faithful projective modules over Noetherian rings are quasi-primaryful modules. In particular, if [Formula: see text] is an Artinian ring, then all [Formula: see text]-modules are quasi-primaryful and the converse is also true when [Formula: see text] is a Noetherian ring.

scholarly journals Computing metric dimension of compressed zero divisor graphs associated to rings

2018 ◽  
Vol 10 (2) ◽  
pp. 298-318
Author(s):  
S. Pirzada ◽  
M. Imran Bhat
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
The Family ◽  
Vertex Set ◽  
Relationship Of ◽  
The Relationship

Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

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