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.

KERNEL IDEALS IN ALMOST DISTRIBUTIVE LATTICES

2012 ◽  
Vol 05 (02) ◽  
pp. 1250024
Author(s):  
M. Sambasiva Rao ◽  
G. C. Rao
Keyword(s):  
Distributive Lattice ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kernel Ideal ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

The concepts of ⋆-congruences and kernel ideals are introduced in a pseudo-complemented almost distributive lattice (ADL). A necessary and sufficient condition for a congruence to become a ⋆-congruence is derived. Some equivalent conditions for every ideal of a pseudo-complemented ADL to become a kernel ideal are derived. Prime kernel ideals are characterized in terms of minimal prime ideals. Properties of kernel ideals are observed in Stone ADLs.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

scholarly journals On the relation of a distributive lattice to its lattice of ideals

1972 ◽  
Vol 7 (3) ◽  
pp. 377-385 ◽  
Author(s):  
Herbert S. Gaskill
Keyword(s):  
Distributive Lattice ◽  
Related Result ◽  
Distributive Lattices ◽  
Relationship Of ◽  
The Relationship ◽  
Finite Distributive Lattices ◽  
Lattice Of Ideals

In this note we examine the relationship of a distributive lattice to its lattice of ideals. Our main result is that a distributive lattice and its lattice of ideals share exactly the same collection of finite sublattices. In addition we give a related result characterizing those finite distributive lattices L which can be embedded in a lattice L′ whenever they can be embedded in its lattice of ideals T(L′).

Minimal prime ideals and arithmetic comprehension

1991 ◽  
Vol 56 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Kostas Hatzikiriakou
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Order Arithmetic ◽  
Minimal Prime ◽  
Weak König’S Lemma ◽  
König’S Lemma

We assume that the reader is familiar with the program of “reverse mathematics” and the development of countable algebra in subsystems of second order arithmetic. The subsystems we are using in this paper are RCA0, WKL0 and ACA0. (The reader who wants to learn about them should study [1].) In [1] it was shown that the statement “Every countable commutative ring has a prime ideal” is equivalent to Weak Konig's Lemma over RCA0, while the statement “Every countable commutative ring has a maximal ideal” is equivalent to Arithmetic Comprehension over RCA0. Our main result in this paper is that the statement “Every countable commutative ring has a minimal prime ideal” is equivalent to Arithmetic Comprehension over RCA0. Minimal prime ideals play an important role in the study of countable commutative rings; see [2, pp. 1–7].

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].)

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.

TOPOLOGICAL CHARACTERIZATION OF DUALLY NORMAL ALMOST DISTRIBUTIVE LATTICES

2012 ◽  
Vol 05 (03) ◽  
pp. 1250043
Author(s):  
G. C. Rao ◽  
N. Rafi ◽  
Ravi Kumar Bandaru
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Topological Characterization ◽  
Maximal Ideals ◽  
Necessary And Sufficient

A dually normal almost distributive lattice is characterized topologically in terms of its maximal ideals and prime ideals. Some necessary and sufficient conditions for the space of maximal ideals to be dually normal are obtained.

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