scholarly journals IDEALS OF POLYNOMIAL SEMIRINGS IN TROPICAL MATHEMATICS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250143
Author(s):  
ZUR IZHAKIAN ◽  
LOUIS ROWEN
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Hilbert Basis ◽  
Basis Theorem ◽  
Tropical Mathematics

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

scholarly journals On Graded 2-Prime Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 493
Author(s):  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Principal Ideal Domain ◽  
Graded Rings ◽  
Graded Ring ◽  
Ideal Domain ◽  
Primary Ideals ◽  
Weakly Prime Ideals

The purpose of this paper is to introduce the concept of graded 2-prime ideals as a new generalization of graded prime ideals. We show that graded 2-prime ideals and graded semi-prime ideals are different. Furthermore, we show that graded 2-prime ideals and graded weakly prime ideals are also different. Several properties of graded 2-prime ideals are investigated. We study graded rings in which every graded 2-prime ideal is graded prime, we call such a graded ring a graded 2-P-ring. Moreover, we introduce the concept of graded semi-primary ideals, and show that graded 2-prime ideals and graded semi-primary ideals are different concepts. In fact, we show that graded semi-primary, graded 2-prime and graded primary ideals are equivalent over Z-graded principal ideal domain.

scholarly journals On weakly S-prime ideals of commutative rings

2021 ◽  
Vol 29 (2) ◽  
pp. 173-186
Author(s):  
Fuad Ali Ahmed Almahdi ◽  
El Mehdi Bouba ◽  
Mohammed Tamekkante
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
New Class ◽  
Weakly Prime Ideals

Abstract Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

Noetherian Rings in Which Every Ideal is a Product of Primary Ideals

1980 ◽  
Vol 23 (4) ◽  
pp. 457-459 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Number Theory ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Primary Ideals

The classical rings of number theory, Dedekind domains, are characterized by the property that every ideal is a product of prime ideals. More generally, a commutative ring R with identity has the property that every ideal is a product of prime ideals if and only if R is a finite direct sum of Dedekind domains and special principal ideal rings. These rings, called general Z.P.I. rings, are also characterized by the property that every (prime) ideal is finitely generated and locally principal.

scholarly journals N Sequence Prime Ideals

2021 ◽  
pp. 3672-3678
Author(s):  
Hemin A. Ahmad ◽  
Parween A. Hummadi
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Closed Ideal ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Strongly Irreducible

    In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

scholarly journals A note on the invertible ideal theorem

1984 ◽  
Vol 25 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Simple Proof ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Central Element ◽  
Commutative Case ◽  
Additional Information ◽  
The Subject

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

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)

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Radicals of PID's and Dedekind Domains

1972 ◽  
Vol 24 (4) ◽  
pp. 566-572 ◽  
Author(s):  
R. E. Propes
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Radical Class ◽  
Dedekind Domains ◽  
Radical Ideals ◽  
The One ◽  
Image Position ◽  
Principal Ideal Domains

The purpose of this paper is to characterize the radical ideals of principal ideal domains and Dedekind domains. We show that if T is a radical class and R is a PID, then T(R) is an intersection of prime ideals of R. More specifically, ifthen T(R) = (p1p2 … pk), where p1, p2, … , pk are distinct primes, and where (p1p2 … Pk) denotes the principal ideal of R generated by p1p2 … pk. We also characterize the radical ideals of commutative principal ideal rings. For radical ideals of Dedekind domains we obtain a characterization similar to the one given for PID's.

scholarly journals L-Fuzzy Semiprime Ideals of a Poset

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Berhanu Assaye Alaba ◽  
Derso Abeje Engidaw
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Semiprime Ideal

In this paper, we introduce the concept of L-fuzzy semiprime ideal in a general poset. Characterizations of L-fuzzy semiprime ideals in posets as well as characterizations of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are obtained. Also, L-fuzzy prime ideals in a poset are characterized.

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

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