scholarly journals Ring Extensions with Finitely Many Non-Artinian Intermediate Rings

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Noômen Jarboui ◽  
Naseam Al-Kuleab ◽  
Omar Almallah
Keyword(s):  
Commutative Ring ◽  
Initial Step ◽  
Intermediate Ring ◽  
Ring Extensions

The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

Polynomial and power series ring extensions from sequences

2020 ◽  
pp. 2250048
Author(s):  
Gyu Whan Chang ◽  
Phan Thanh Toan
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Trivial Case ◽  
Addition Formula ◽  
Power Series Ring ◽  
Series Ring ◽  
Ring Extensions ◽  
General Sequences ◽  
Divisibility Properties ◽  
Positive Integers

Let [Formula: see text] be a commutative ring with identity. Let [Formula: see text] and [Formula: see text] be the collection of polynomials and, respectively, of power series with coefficients in [Formula: see text]. There are a lot of multiplications in [Formula: see text] and [Formula: see text] such that together with the usual addition, [Formula: see text] and [Formula: see text] become rings that contain [Formula: see text] as a subring. These multiplications are from a class of sequences [Formula: see text] of positive integers. The trivial case of [Formula: see text], i.e. [Formula: see text] for all [Formula: see text], gives the usual polynomial and power series ring. The case [Formula: see text] for all [Formula: see text] gives the well-known Hurwitz polynomial and Hurwitz power series ring. In this paper, we study divisibility properties of these polynomial and power series ring extensions for general sequences [Formula: see text] including UFDs and GCD-domains. We characterize when these polynomial and power series ring extensions are isomorphic to each other. The relation between them and the usual polynomial and power series ring is also presented.

scholarly journals Primitiewe elemente vir kommutatiewe ringuitbreidings

1991 ◽  
Vol 10 (2) ◽  
pp. 67-71
Author(s):  
H. J. Schutte
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Integral Domain ◽  
Prime Ideals ◽  
Primitive Elements ◽  
Integral Domains ◽  
Field Extensions ◽  
Ring Extensions ◽  
Minimal Prime

The existence of primitive elements for integral domain extensions is considered with reference to the well known theorem about primitive elements for field extensions. Primitive elements for extensions of a commutative ring R with identity are considered, where R has only a finite number of minimal prime ideals with zero intersection. This case is reduced to the case for ring extensions of integral domains.

scholarly journals On 1-absorbing δ-primary ideals

2021 ◽  
Vol 29 (3) ◽  
pp. 135-150
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou ◽  
Ünsal Tekir ◽  
Suat Koç
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Basic Properties ◽  
New Class ◽  
Ring Extensions ◽  
Primary Ideals ◽  
Nonzero Identity

Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

On almost valuation ring pairs

2020 ◽  
pp. 2150182
Author(s):  
Noômen Jarboui ◽  
David E. Dobbs
Keyword(s):  
Positive Characteristic ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Text Type ◽  
Intermediate Ring ◽  
Valuation Domains ◽  
Divided Domain ◽  
Integrally Closed ◽  
Ring Extensions ◽  
Ring Pair

If [Formula: see text] are (commutative) rings, [Formula: see text] denotes the set of intermediate rings and [Formula: see text] is called an almost valuation (AV)-ring pair if each element of [Formula: see text] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let [Formula: see text] be rings, with [Formula: see text] denoting the integral closure of [Formula: see text] in [Formula: see text]. Then [Formula: see text] is an AV-ring pair if and only if both [Formula: see text] and [Formula: see text] are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions [Formula: see text]. If [Formula: see text] is an AV-ring pair, then [Formula: see text] is a P-extension. The AV-ring pairs [Formula: see text] arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of [Formula: see text] type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for [Formula: see text] being an AV-ring pair to entail that [Formula: see text] is an overring of [Formula: see text], but there exist domain-theoretic counter-examples to such a conclusion in general. If [Formula: see text] is an AV-ring pair and [Formula: see text] satisfies FCP, then each intermediate ring either contains or is contained in [Formula: see text]. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

scholarly journals On q-Galois Extensions of Simple Rings

1966 ◽  
Vol 27 (2) ◽  
pp. 485-507 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Galois Theory ◽  
Galois Extensions ◽  
Late Professor ◽  
Simple Ring ◽  
Infinite Dimensional ◽  
Intermediate Ring ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Ring Extensions

In 1952, the late Professor T. Nakayama succeeded in constructing the Galois theory for finite dimensional simple ring extensions [7]. And, we believe, the theory was essentially due to the following proposition: If a simple ring A is Galois and finite over a simple subring B then A is B′-A-completely reducible for any simple intermediate ring B′ of A/B [7, Lemmas 1.1 and 1.2]. Moreover, as was established in [5], Nakayama’s idea was still efficient in considering the infinite dimensional Galois theory of simple rings.

Unit groups and commutative ring extensions

1988 ◽  
Vol 16 (11) ◽  
pp. 2349-2361
Author(s):  
Jan Krempa
Keyword(s):  
Commutative Ring ◽  
Ring Extensions
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