WHEN IS THE INTEGRAL CLOSURE COMPARABLE TO ALL INTERMEDIATE RINGS

2016 ◽  
Vol 95 (1) ◽  
pp. 14-21 ◽  
Author(s):  
MABROUK BEN NASR ◽  
NABIL ZEIDI

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].

Author(s):  
Ahmed Ayache

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.


1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler

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.


2019 ◽  
Vol 18 (06) ◽  
pp. 1950104 ◽  
Author(s):  
Najib Ouled Azaiez ◽  
Moutu Abdou Salam Moutui

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.


2019 ◽  
Vol 73 (1) ◽  
pp. 1-8
Author(s):  
Lhoussain El Fadil

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.


2011 ◽  
Vol 10 (06) ◽  
pp. 1343-1350
Author(s):  
MOHAMMED KABBOUR ◽  
NAJIB MAHDOU

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation A ⋈f J. We provide necessary and sufficient conditions for A ⋈f J to be an elementary divisor ring where A and B are integral domains. In this case it is shown that A ⋈f J is an Hermite ring if and only if it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring A along an A-submodule E of Q(A) such that E2 ⊆ E.


2017 ◽  
Vol 5 (2) ◽  
pp. 117 ◽  
Author(s):  
Jayanta Ghosh ◽  
Dhananjoy Mandal ◽  
Tapas Samanta

Concept of soft equivalence relations (classes, mappings) are introduced using the notion of soft elements. Then we redefine the notion of soft group and soft ring in a new way by using the idea of soft elements and it is seen that our definitions of soft group and soft ring are equivalent to the existing notions of soft group [2] and soft ring [1]. The notion of soft coset is presented and validated by suitable examples. We investigate some important properties like soft divisor of zero, characteristic of a soft ring etc. by considering examples. Moreover, some necessary and sufficient conditions are established for a soft ring to be a soft integral domain and soft field.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Subramanian Visweswaran

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLet S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.Originality/valueMotivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.


1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell

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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