scholarly journals Cardinalities of prime spectra of precompletions

2021 ◽  
pp. 133-152
Author(s):  
Erica Barrett ◽  
Emil Graf ◽  
S. Loepp ◽  
Kimball Strong ◽  
Sharon Zhang
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Local Domain ◽  
Content Type ◽  
Necessary And Sufficient

Given a complete local (Noetherian) ring T T , we find necessary and sufficient conditions on T T such that there exists a local domain A A with | A | > | T | |A| > |T| and A ^ = T \widehat {A} = T , where A ^ \widehat {A} denotes the completion of A A with respect to its maximal ideal. We then find necessary and sufficient conditions on T T such that there exists a domain A A with A ^ = T \widehat {A} = T and | S p e c ( A ) | > | S p e c ( T ) | |\mathrm {Spec}(A)| > |\mathrm {Spec}(T)| . Finally, we use “partial completions” to create local rings A A with A ^ = T \widehat {A} = T such that S p e c ( A ) \mathrm {Spec}(A) has varying cardinality in different varieties.

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

1972 ◽  
Vol 24 (4) ◽  
pp. 703-712 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Torsion Class ◽  
Ring Of Quotients ◽  
Ore Condition ◽  
The Right ◽  
Necessary And Sufficient

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

Formal Fibers of Unique Factorization Domains

2010 ◽  
Vol 62 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Adam Boocher ◽  
Michael Daub ◽  
Ryan K. Johnson ◽  
H. Lindo ◽  
S. Loepp ◽  
...  
Keyword(s):  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Tight Closure ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient ◽  
Unique Factorization Domains ◽  
Adic Completion

AbstractLet (T,M) be a complete local (Noetherian) ring such that dimT ≥ 2 and |T| = |T/M| and let ﹛pi﹜i∈𝒥 be a collection of elements of T indexed by a set I so that |𝒥| < |T|. For each i ∈ 𝒥, let Ci := ﹛Qi1, … ,Qini ﹜ be a set of nonmaximal prime ideals containing pi such that the Qi j are incomparable and pi ∈ Qjk if and only if i = j. We provide necessary and sufficient conditions so that T is the m -adic completion of a local unique factorization domain (A,m ), and for each i ∈ I, there exists a unit ti of T so that pi ti ∈ A andCi is the set of prime ideals Q of T that are maximal with respect to the condition that Q ∩ A = pi tiA.We then use this result to construct a (nonexcellent) unique factorization domain containingmany ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

About a Serre-type condition for modules

2017 ◽  
Vol 16 (11) ◽  
pp. 1750206 ◽  
Author(s):  
Samaneh Tabejamaat ◽  
Amir Mafi
Keyword(s):  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Type Condition ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a commutative Noetherian ring with nonzero identity and [Formula: see text] be a finitely generated [Formula: see text]-module. This note gives some necessary and sufficient conditions of a module to satisfy the Serre-type condition [Formula: see text].

Subprojectivity domains of pure-projective modules

2019 ◽  
Vol 19 (05) ◽  
pp. 2050091
Author(s):  
Yılmaz Durğun
Keyword(s):  
Noetherian Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Projective Modules ◽  
Finitely Generated ◽  
Flat Modules ◽  
Necessary And Sufficient

In a recent paper, Holston et al. have defined a module [Formula: see text] to be [Formula: see text]-subprojective if for every epimorphism [Formula: see text] and homomorphism [Formula: see text], there exists a homomorphism [Formula: see text] such that [Formula: see text]. Clearly, every module is subprojective relative to any projective module. For a module [Formula: see text], the subprojectivity domain of [Formula: see text] is defined to be the collection of all modules [Formula: see text] such that [Formula: see text] is [Formula: see text]-subprojective. We consider, for every pure-projective module [Formula: see text], the subprojective domain of [Formula: see text]. We show that the flat modules are the only ones sharing the distinction of being in every single subprojectivity domain of pure-projective modules. Pure-projective modules whose subprojectivity domain is as small as possible will be called pure-projective indigent (pp-indigent). Properties of subprojectivity domains of pure-projective modules and of pp-indigent modules are studied. For various classes of modules (such as simple, cyclic, finitely generated and singular), necessary and sufficient conditions for the existence of pp-indigent modules of those types are studied. We characterize the structure of a Noetherian ring over which every (simple, cyclic, finitely generated) pure-projective module is projective or pp-indigent. Furthermore, finitely generated pp-indigent modules on commutative Noetherian hereditary rings are characterized.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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