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].)

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).

Zariski Topologies on Graded Ideals

2021 ◽  
Vol 78 (1) ◽  
pp. 215-224
Author(s):  
Malik Bataineh ◽  
Azzh Saad Alshehry ◽  
Rashid Abu-Dawwas
Keyword(s):  
Commutative Ring ◽  
Zariski Topology ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Algebraic Properties

Abstract In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

TWO SPACES OF MINIMAL PRIMES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250014 ◽  
Author(s):  
PAPIYA BHATTACHARJEE
Keyword(s):  
Compact Space ◽  
Hausdorff Space ◽  
Topological Spaces ◽  
Zariski Topology ◽  
Topological Properties ◽  
Locally Compact ◽  
Inverse Topology ◽  
Minimal Prime ◽  
Stone Spaces ◽  
Prime Elements

This paper studies algebraic frames L and the set Min (L) of minimal prime elements of L. We will endow the set Min (L) with two well-known topologies, known as the Hull-kernel (or Zariski) topology and the inverse topology, and discuss several properties of these two spaces. It will be shown that Min (L) endowed with the Hull-kernel topology is a zero-dimensional, Hausdorff space; whereas, Min (L) endowed with the inverse topology is a T1, compact space. The main goal will be to find conditions on L for the spaces Min (L) and Min (L)-1 to have various topological properties; for example, compact, locally compact, Hausdorff, zero-dimensional, and extremally disconnected. We will also discuss when the two topological spaces are Boolean and Stone spaces.

Coloring of lattices

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
S. Nimbhokar ◽  
M. Wasadikar ◽  
M. Pawar
Keyword(s):  
Distributive Lattice ◽  
Chromatic Number ◽  
Finite Lattice ◽  
Clique Number ◽  
Prime Ideals ◽  
Minimal Prime

AbstractThe concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.

Reticulation of an almost distributive lattice

2015 ◽  
Vol 08 (04) ◽  
pp. 1550077
Author(s):  
Y. S. Pawar ◽  
I. A. Shaikh
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Prime Spectrum ◽  
Minimal Prime

A congruence relation [Formula: see text] on an ADL [Formula: see text] is defined such that the quotient lattice [Formula: see text] is a distributive lattice and the prime spectrum of [Formula: see text] and of [Formula: see text] are homeomorphic. Also it is proved that the minimal prime spectrum of [Formula: see text] is homeomorphic with the minimal prime spectrum of [Formula: see text].

ANNULETS AND α-IDEALS OF C-ALGEBRAS

2013 ◽  
Vol 06 (04) ◽  
pp. 1350060
Author(s):  
M. Sambasiva Rao
Keyword(s):  
Prime Ideals ◽  
Topological Properties ◽  
C Algebra ◽  
C Algebras ◽  
Minimal Prime ◽  
Equivalent Conditions

The concepts of annulets and α-ideals are introduced in C-algebras. α-ideals are characterized in terms of annulets and minimal prime ideals. A set of equivalent conditions is established for every ideal of a C-algebra to become an α-ideal. Some topological properties of the class of all prime α-ideals are studied in a C-algebra.

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 Spaces of ideals of distributive lattices II. Minimal prime ideals

1974 ◽  
Vol 18 (1) ◽  
pp. 54-72 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Distributive Lattice ◽  
Commutative Rings ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Commutative Semigroups ◽  
Satisfactory State ◽  
Minimal Prime ◽  
The Relationship

This paper, the second of a sequence beginning with [14], deals with the relationship between a distributive lattice L = (L; ∨, ∧, 0) with zero, and certain spaces of minimal prime ideals of L. Similar studies of minimal prime ideals in commutative semigroups [8] and in commutative rings [6] inspired this work, and many of our results are similar to ones in these two articles. However the nature of our situation enables many of these results to be pushed deeper and thus to arrive at a more satisfactory state; indeed with the insight obtained from the simpler lattice situation, one can return to some topics considered in [6], [8] and give complete accounts. We do not do this in the present paper, but leave the details to the reader, see e.g. [15]. Also a study of minimal prime ideals illuminates some topics in the theory of distributive lattices, particularly Stone lattices.

Maximal d-Ideals in a Riesz Space

1983 ◽  
Vol 35 (6) ◽  
pp. 1010-1029 ◽  
Author(s):  
Charles B. Huijsmans ◽  
Ben de Pagter
Keyword(s):  
Riesz Space ◽  
Weak Order ◽  
Prime Ideals ◽  
Topological Properties ◽  
Order Unit ◽  
Structure Space ◽  
Maximal Ideals ◽  
Minimal Prime ◽  
The Ideal

We recall that the ideal I in an Archimedean Riesz space L is called a d-ideal whenever it follows from ƒ ∊ I that {ƒ}dd ⊂ I. Several authors (see [4], [5], [6], [12], [13], [15] and [18]) have considered the class of all d-ideals in L, but the set ℐd of all maximal d-ideals in L has not been studied in detail in the literature. In [12] and [13] the present authors paid some attention to certain aspects of the theory of maximal d-ideals, however neglecting the fact thatℐd, equipped with its hull-kernel topology, is a structure space of the underlying Riesz space L.The main purpose of the present paper is to investigate the topological properties of ℐd and to compare ℐd to other structure spaces of L, such as the space of minimal prime ideals and the space of all e-maximal ideals in L (where e > 0 is a weak order unit).

scholarly journals On $n$-Absorbing Ideals in a Lattice

2021 ◽  
Vol 45 (4) ◽  
pp. 597-605
Author(s):  
ALI AKBAR ESTAJI ◽  
◽  
TOKTAM HAGHDADI ◽  
Keyword(s):  
Positive Integer ◽  
Distributive Lattice ◽  
Prime Ideals ◽  
Minimal Prime

Let L be a lattice, and let n be a positive integer. In this article, we introduce n-absorbing ideals in L. We give some properties of such ideals. We show that every n-absorbing ideal I of L has at most n minimal prime ideals. Also, we give some properties of 2-absorbing and weakly 2-absorbing ideals in L. In particular we show that in every non-zero distributive lattice L, 2-absorbing and weakly 2-absorbing ideals are equivalent.

Sign in / Sign up

Export Citation Format

Share Document