scholarly journals Ideals on neutrosophic extended triplet groups

2021 ◽  
Vol 7 (3) ◽  
pp. 4767-4777
Author(s):  
Xin Zhou ◽  
◽  
Xiao Long Xin ◽  
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Topological Structure ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Ideal Space ◽  
Necessary And Sufficient

<abstract><p>In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $ T_{1} $-space and a Hausdorff space. </p></abstract>

PRIME IDEAL CHARACTERIZATION OF STONE ADLs

2010 ◽  
Vol 03 (02) ◽  
pp. 357-367 ◽  
Author(s):  
U. M. Swamy ◽  
S. Ramesh ◽  
Ch. Shanthi Sundar Raj
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper we obtain certain necessary and sufficient conditions for an almost distributive lattice to become a Stone almost distributive lattice in topological and algebraic terms.

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.

scholarly journals On generalizations of quasi-prime ideals of an ordered left almost semigroups

2021 ◽  
Author(s):  
Pairote Yiarayong
Keyword(s):  
Prime Ideal ◽  
Left Ideal ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Necessary And Sufficient

AbstractThe purposes of this paper are to introduce generalizations of quasi-prime ideals to the context of $$\phi $$ ϕ -quasi-prime ideals. Let $$\phi : {\mathcal {I}}(S) \rightarrow {\mathcal {I}}(S) \cup \left\{ \emptyset \right\} $$ ϕ : I ( S ) → I ( S ) ∪ ∅ be a function where $$ {\mathcal {I}}(S)$$ I ( S ) is the set of all left ideals of an ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroup S. A proper left ideal A of an ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroup S is called a $$\phi $$ ϕ -quasi-prime ideal, if for each $$a, b\in S$$ a , b ∈ S with $$ab \in A - \phi (A)$$ a b ∈ A - ϕ ( A ) , then $$a \in A$$ a ∈ A or $$b\in A$$ b ∈ A . Some characterizations of quasi-prime and $$\phi $$ ϕ -quasi-prime ideals are obtained. Moreover, we investigate relationships between weakly quasi-prime, almost quasi-prime, $$\omega $$ ω -quasi-prime, m-quasi-prime and $$\phi $$ ϕ -quasi-prime ideals of ordered $${{\mathcal {L}}}{{\mathcal {A}}}$$ L A -semigroups. Finally, we obtain necessary and sufficient conditions of $$\phi $$ ϕ -quasi-prime ideal in order to be a quasi-prime ideal.

scholarly journals Subdirect products of semirings

2001 ◽  
Vol 26 (9) ◽  
pp. 539-545
Author(s):  
P. Mukhopadhyay
Keyword(s):  
Distributive Lattice ◽  
Subdirect Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Subdirect Products ◽  
Necessary And Sufficient

Bandelt and Petrich (1982) proved that an inversive semiringSis a subdirect product of a distributive lattice and a ring if and only ifSsatisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the “ring” involved can be gradually enriched to a “field.” Finally, we provide a construction of fullE-inversive semirings, which are subdirect products of a semilattice and a ring.

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.

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).

On Countably Paracompact Normal Spaces

1957 ◽  
Vol 9 ◽  
pp. 443-449 ◽  
Author(s):  
M. J. Mansfield
Keyword(s):  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Space ◽  
Countably Paracompact ◽  
Necessary And Sufficient

A. H. Stone (9), E. Michael (3, 4), J. L. Kelley and J. S. Griff en (2) have established many necessary and sufficient conditions that a regular Hausdorff space be paracompact. It is the purpose of this note to show that if the word “countable” is inserted in the appropriate places in the above-mentioned conditions they become, in general, necessary and sufficient conditions that a normal space be countably paracompact.

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

A theorem of Stone-Weierstrass type

1966 ◽  
Vol 62 (4) ◽  
pp. 649-666 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Weierstrass Theorem ◽  
Necessary And Sufficient ◽  
Complex Valued

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

scholarly journals The distribution of prime ideals of a Dedekind domain

1974 ◽  
Vol 11 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Anne P. Grams
Keyword(s):  
Class Group ◽  
Sufficient Conditions ◽  
Torsion Group ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Class Groups ◽  
Maximal Ideals ◽  
The Family ◽  
Necessary And Sufficient

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.

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