scholarly journals Relative Zariski Open Objects

2012 ◽  
Vol 10 (1) ◽  
pp. 9-39 ◽  
Author(s):  
Florian Marty
Keyword(s):  
Topological Space ◽  
Monoidal Category ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Scheme Theory ◽  
Usual Case ◽  
Definition Of ◽  
Affine Scheme

AbstractIn [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category (,⊗, 1) One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoidal objects in . The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings (ℤ-mod,⊗,ℤ). The main result states that for any commutative monoidal object A in , the locale of Zariski open subobjects of the affine scheme Spec(A) is associated to a topological space whose points are prime ideals of A and whose open subsets are defined by the same formula as in rings. As a consequence, we can compare the notions of scheme over in [D] and in [TV].

CPI-Extensions: Overrings of Integral Domains with Special Prime Spectrums

1977 ◽  
Vol 29 (4) ◽  
pp. 722-737 ◽  
Author(s):  
Monte B. Boisen ◽  
Philip B. Sheldon
Keyword(s):  
Topological Space ◽  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Integral Domains ◽  
Partially Ordered ◽  
Set Inclusion

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

scholarly journals Zariski topology on the spectrum of graded pseudo prime submodules

2021 ◽  
Vol 39 (3) ◽  
pp. 17-26
Author(s):  
Rashid Abu-Dawwas
Keyword(s):  
Topological Space ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Graded Modules ◽  
Prime Submodules ◽  
Algebraic Properties

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.

The Minimal Prime Spectrum of a Commutative Ring

1971 ◽  
Vol 23 (5) ◽  
pp. 749-758 ◽  
Author(s):  
M. Hochster
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Topological Characterization ◽  
Completely Regular ◽  
Minimal Prime

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

scholarly journals On the gamma spectrum of multiplication gamma acts

2021 ◽  
Vol 48 (2) ◽  
Author(s):  
Mehdi S. Abbas ◽  
◽  
Samer A. Gubeir ◽  
Keyword(s):  
Topological Space ◽  
Gamma Spectrum ◽  
Zariski Topology ◽  
Topological Properties ◽  
Separation Axioms ◽  
Algebraic Properties

In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topological properties of this topology are studied. Various algebraic properties of topological gamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.

Radical Regularity in Differential Rings

1971 ◽  
Vol 23 (2) ◽  
pp. 197-201 ◽  
Author(s):  
Howard E. Gorman
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Prime Ideals ◽  
Differential Ring ◽  
Quotient Maps ◽  
The Stability ◽  
Definition Of ◽  
Finitely Generated Modules ◽  
Regular Rings ◽  
The Ideal

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

Notes on mildly distributive semilattices

2017 ◽  
Vol 67 (5) ◽  
Author(s):  
Sergio Arturo Celani ◽  
Luciano Javier González
Keyword(s):  
Prime Ideals ◽  
Distributive Semilattice ◽  
Definition Of ◽  
Distributive Semilattices ◽  
Distributive Extension

AbstractIn this paper we shall investigate the mildly distributive meet-semilattices by means of the study of their filters and Frink-ideals as well as applying the theory of annihilator. We recall some characterizations of the condition of mildly-distributivity and we give several new characterizations. We prove that the definition of strong free distributive extension, introduced by Hickman in 1984, can be simplified and we show a correspondence between (prime) Frink-ideals of a mildly distributive semilattice and (prime) ideals of its strong free distributive extension.

Asymptotic Behavior of Integral Closures in Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 505-514 ◽  
Author(s):  
R. Naghipour ◽  
P. Schenzel
Keyword(s):  
Asymptotic Behavior ◽  
Integral Closure ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Large N ◽  
Nagata Ring ◽  
Associated Prime Ideals ◽  
Integral Closures ◽  
Definition Of ◽  
Associated Prime

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

A localic approach to minimal prime spectra

1988 ◽  
Vol 103 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Sun Shu-Hao
Keyword(s):  
Distributive Lattice ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Minimal Prime

Throughout this paper, A will denote a distributive lattice with 0 and 1; we shall write spec A for the prime spectrum of A (i.e. the set of prime ideals of A, with the Stone–Zariski topology), and max A, min A for the subspaces of spec A consisting of maximal and minimal prime ideals respectively. These two subspaces have rather different topological properties: max A is always compact, but not always Hausdorff (indeed, any compact T1-space can occur as max A for some A), and min A is always Hausdorff (in fact zero-dimensional), but not always compact. (For more information on max A and min A, see Simmons[3].)

scholarly journals Objets compacts dans les topos

1986 ◽  
Vol 40 (2) ◽  
pp. 203-217 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Jacques Penon
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Compact Manifolds ◽  
Topological Spaces ◽  
Internal Logic ◽  
Definition Of

AbstractIt is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

scholarly journals The Logical Complexity of Finitely Generated Commutative Rings

2018 ◽  
Vol 2020 (1) ◽  
pp. 112-166 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Anatole Khélif ◽  
Eudes Naziazeno ◽  
Thomas Scanlon
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Dual Numbers ◽  
Finitely Generated ◽  
Nonstandard Model

AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.

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