scholarly journals Dedekind’s Criterion and Integral Bases

2019 ◽  
Vol 73 (1) ◽  
pp. 1-8
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Principal Ideal Domain ◽  
Quotient Field ◽  
Ideal Domain ◽  
Integral Bases ◽  
Necessary And Sufficient

Abstract Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤL: R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

scholarly journals On solvability of the matrix equation AXB = C over a principal ideal domain

2020 ◽  
pp. 47-50
Author(s):  
Volodymyr Prokip
Keyword(s):  
Matrix Equation ◽  
Principal Ideal ◽  
Normal Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
The Matrix ◽  
Smith Normal Forms ◽  
Necessary And Sufficient

In this paper we present conditions of solvability of the matrix equation AXB = B over a principal ideal domain. The necessary and sufficient conditions of solvability of equation AXB = B in term of the Smith normal forms and in term of the Hermi-te normal forms of matrices constructed in a certain way by using the coefficients of this equation are proposed. If a solution of this equation exists we propose the method for its construction.

Arithmetical Semigroup Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
Bonnie R. Hardy ◽  
Thomas S. Shores
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Forthcoming Paper ◽  
Semigroup Ring ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Rings ◽  
Principal Ideal Ring ◽  
Arithmetical Semigroup ◽  
Necessary And Sufficient

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

Almost valuation property in bi-amalgamations and pairs of rings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950104 ◽  
Author(s):  
Najib Ouled Azaiez ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Ring Extensions ◽  
Necessary And Sufficient

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

WHEN IS THE INTEGRAL CLOSURE COMPARABLE TO ALL INTERMEDIATE RINGS

2016 ◽  
Vol 95 (1) ◽  
pp. 14-21 ◽  
Author(s):  
MABROUK BEN NASR ◽  
NABIL ZEIDI
Keyword(s):  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Integral Domains ◽  
Necessary And Sufficient ◽  
Quotient Rings

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

Twisted Brauer monoids

2018 ◽  
Vol 148 (4) ◽  
pp. 731-750 ◽  
Author(s):  
Igor Dolinka ◽  
James East
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Singular Part ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Generating Set ◽  
Idempotent Rank ◽  
Singular Ideal ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals

We investigate the structure of the twisted Brauer monoid , comparing and contrasting it with the structure of the (untwisted) Brauer monoid . We characterize Green's relations and pre-orders on , describe the lattice of ideals and give necessary and sufficient conditions for an ideal to be idempotent generated. We obtain formulae for the rank (smallest size of a generating set) and (where applicable) the idempotent rank (smallest size of an idempotent generating set) of each principal ideal; in particular, when an ideal is idempotent generated, its rank and idempotent rank are equal. As an application of our results, we describe the idempotent generated subsemigroup of (which is not an ideal), as well as the singular ideal of (which is neither principal nor idempotent generated), and we deduce that the singular part of the Brauer monoid is idempotent generated, a result previously proved by Maltcev and Mazorchuk.

scholarly journals FINITENESS CONDITIONS OF S-COHN–JORDAN EXTENSIONS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250063
Author(s):  
JERZY MATCZUK
Keyword(s):  
Representation Theory ◽  
Principal Ideal ◽  
Sufficient Conditions ◽  
Ring Theory ◽  
Necessary And Sufficient Conditions ◽  
Finiteness Conditions ◽  
Principal Ideal Ring ◽  
Ideal Ring ◽  
Necessary And Sufficient

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn–Jordan extension of R. Some results relating finiteness conditions of R and that of A(R; S) are presented. In particular necessary and sufficient conditions for A(R; S) to be left noetherian, to be left Bézout and to be left principal ideal ring are presented. This also offers a solution to Problem 10 from [On S-Cohn–Jordan extensions, in Proc. 39th Symp. Ring Theory and Representation Theory, Hiroshima, ed. M. Kutami (Hiroshima Univ., Japan, 2007), pp. 30–35].

scholarly journals B-almost distributive fuzzy lattice

2018 ◽  
Vol 47 (3) ◽  
Author(s):  
Berhanu Assaye ◽  
Mihret Alemneh ◽  
Gerima Tefera
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fuzzy Algebra ◽  
Fuzzy Lattice ◽  
Necessary And Sufficient

The paper introduces the concept of B-Almost distributive fuzzy lattice (BADFL) in terms of its principal ideal fuzzy lattice. Necessary and sufficient conditions for an ADFL to become a B-ADFL are investigated. We also prove the equivalency of B-algebra and B-fuzzy algebra. In addition, we extend PSADL to PSADFL and prove that B-ADFL implies PSADFL.

scholarly journals Integral Bases in Kummer Extensions of Dedekind Fields

1963 ◽  
Vol 15 ◽  
pp. 755-765 ◽  
Author(s):  
Leon R. McCulloh
Keyword(s):  
Integral Closure ◽  
Sufficient Condition ◽  
Dedekind Ring ◽  
Necessary And Sufficient Condition ◽  
Integral Basis ◽  
Separable Extension ◽  
Quotient Field ◽  
Integral Bases ◽  
Necessary And Sufficient

Let J be a Dedekind ring, F its quotient field, F′ a finite separable extension of F, and J′ the integral closure of J in F′. It has been shown by Artin (1) that a necessary and sufficient condition that J′ have an integral basis over J is that a certain ideal of F (namely, √(D/Δ), where D is the discriminant of the extension and Δ is the discriminant of an arbitrary basis of the extension) should be principal. More generally, he showed that if is an ideal of F′, then a necessary and sufficient condition that have a module basis over J is that N()√(D/Δ) should be principal.

When is a fixed ring comparable to all overrings?

2020 ◽  
pp. 2250050
Author(s):  
Ahmed Ayache
Keyword(s):  
Integral Domain ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Necessary And Sufficient Conditions ◽  
Fixed Ring ◽  
Necessary And Sufficient

An overring [Formula: see text] of an integral domain [Formula: see text] is said to be comparable if [Formula: see text], [Formula: see text], and each overring of [Formula: see text] is comparable to [Formula: see text] under inclusion. We do provide necessary and sufficient conditions for which [Formula: see text] has a comparable overring. Several consequences are derived, specially for minimal overrings, or in the case where the integral closure [Formula: see text] of [Formula: see text] is a comparable overring, or also when each chain of distinct overrings of [Formula: see text] is finite.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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