On maximal ideals which are also minimal prime ideals in certain Banach rings

2020 ◽  
Vol 250 (2) ◽  
pp. 179-209
Author(s):  
Tomoki Mihara
Keyword(s):  
Prime Ideals ◽  
Maximal Ideals ◽  
Minimal Prime

On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

2018 ◽  
Vol 85 (3-4) ◽  
pp. 356
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Integral Closure ◽  
Prime Ideals ◽  
Local Rings ◽  
Finite Sets ◽  
Maximal Ideals ◽  
Valuation Rings ◽  
Canonical Bijection ◽  
Minimal Prime

<p>It is shown that, for all local rings (R,M), there is a canonical bijection between the set <em>DO(R)</em> of depth one minimal prime ideals ω in the completion <em><sup>^</sup>R</em> of <em>R</em> and the set <em>HO(R/Z)</em> of height one maximal ideals <em>̅M'</em> in the integral closure <em>(R/Z)'</em> of <em>R/Z</em>, where <em>Z </em>:<em>= Rad(R)</em>. Moreover, for the finite sets <strong>D</strong> := {<em>V*/V* </em>:<em>= (<sup>^</sup>R/ω)'</em>, ω ∈ DO(R)} and H := {<em>V/V := (R/Z)'<sub><em>̅M'</em></sub>, <em>̅M'</em> ∈ HO(R/Z)</em>}:</p><p>(a) The elements in <strong>D</strong> and <strong>H</strong> are discrete Noetherian valuation rings.</p><p>(b) <strong>D</strong> = {<em><sup>^</sup>V</em> ∈ <strong>H</strong>}.</p>

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 Essential ideals in subrings of C(X) that contain C*(X)

Filomat ◽  
2015 ◽  
Vol 29 (7) ◽  
pp. 1631-1637
Author(s):  
A. Taherifar
Keyword(s):  
Prime Ideals ◽  
Intermediate Ring ◽  
Maximal Ideals ◽  
Minimal Prime

Let A(X) be a subring of C(X) that contains C*(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences ZA and ?A are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C*(X) and C(X), respectively, to any intermediate ring A(X). Moreover, the inverse map Z-1A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, first, we characterize essential ideals in A(X). Afterwards, we show that Z-1A maps essential (resp., free) z-filters on X to essential (resp., free) ideals in A(X) and Z-1A maps essential ?A-filters to essential ideals. Similar to C(X) we observe that the intersection of all essential minimal prime ideals in A(X) is equal to the socle of A(X). Finally, we give a new characterization for the intersection of all essential maximal ideals of A(X).

On minimal prime ideals of commutative Bezout rings

1999 ◽  
Vol 51 (7) ◽  
pp. 1129-1134
Author(s):  
B. V. Zabavskii ◽  
A. I. Gatalevich
Keyword(s):  
Prime Ideals ◽  
Minimal Prime

scholarly journals Characterizations of m-Normal Nearlattices in terms of Principal n-Ideals

2013 ◽  
Vol 38 ◽  
pp. 49-59
Author(s):  
MS Raihan
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Single Element ◽  
Finitely Generated ◽  
Fixed Element ◽  
Minimal Prime

A convex subnearlattice of a nearlattice S containing a fixed element n?S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal n-ideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In this paper, we include several characterizations of those Pn(S) which form m-normal nearlattices. We also show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,…., Pm of S, Po ? … ? Pm = S. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16548 Rajshahi University J. of Sci. 38, 49-59 (2010)

Elements of Minimal Prime Ideals in General Rings

2010 ◽  
pp. 69-81 ◽  
Author(s):  
W. D. Burgess ◽  
A. Lashgari ◽  
A. Mojiri
Keyword(s):  
Prime Ideals ◽  
Minimal Prime

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.

Some properties about the zero-divisor graphs of quasi-ordered sets

2019 ◽  
Vol 19 (04) ◽  
pp. 2050074
Author(s):  
Junye Ma ◽  
Qingguo Li ◽  
Hui Li
Keyword(s):  
Zero Divisor ◽  
Ordered Set ◽  
Line Graph ◽  
Prime Ideals ◽  
Complete Graphs ◽  
Ordered Sets ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Complete Bipartite Graphs ◽  
Minimal Prime

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

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