Overrings of Bezout Domains

1973 ◽  
Vol 16 (4) ◽  
pp. 475-477 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Local Ring ◽  
Left Ideal ◽  
Quotient Ring ◽  
Valuation Domain ◽  
Quotient Field ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Chain Conditions ◽  
Bezout Domains

In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

Blow Up Sequences and the Module of nth Order Differentials

1976 ◽  
Vol 28 (6) ◽  
pp. 1289-1301 ◽  
Author(s):  
William C. Brown
Keyword(s):  
Singular Point ◽  
Local Ring ◽  
Algebraic Curve ◽  
Blow Up ◽  
Quotient Ring ◽  
Integral Closure ◽  
Closed Field ◽  
Quotient Field ◽  
Algebraically Closed Field

Let C denote an irreducible, algebraic curve defined over an algebraically closed field k. Let ? be a singular point of C. We shall employ the following notation throughout the rest of this paper: R will denote the local ring at P, K the quotient field of the integral closure of R in K, A the completion of R with respect to its radical topology, and Ā the integral closure of A in its total quotient ring.

scholarly journals Rings of Fractions and Localization

2020 ◽  
Vol 28 (1) ◽  
pp. 79-87
Author(s):  
Yasushige Watase
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Quotient Ring ◽  
Formal Proof ◽  
Universal Property ◽  
Quotient Field ◽  
Closed Set ◽  
Ring Of Fractions

SummaryThis article formalized rings of fractions in the Mizar system [3], [4]. A construction of the ring of fractions from an integral domain, namely a quotient field was formalized in [7].This article generalizes a construction of fractions to a ring which is commutative and has zero divisor by means of a multiplicatively closed set, say S, by known manner. Constructed ring of fraction is denoted by S~R instead of S−1R appeared in [1], [6]. As an important example we formalize a ring of fractions by a particular multiplicatively closed set, namely R \ p, where p is a prime ideal of R. The resulted local ring is denoted by Rp. In our Mizar article it is coded by R~p as a synonym.This article contains also the formal proof of a universal property of a ring of fractions, the total-quotient ring, a proof of the equivalence between the total-quotient ring and the quotient field of an integral domain.

scholarly journals Notes on topological rings

2013 ◽  
Vol 29 (2) ◽  
pp. 267-273
Author(s):  
MIHAIL URSUL ◽  
◽  
MARTIN JURAS ◽  
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Compact Ring ◽  
Ring Topology ◽  
Bezout Domain ◽  
Totally Bounded ◽  
Ideal Domain ◽  
Nilpotent Ring ◽  
Trivial Multiplication

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

A Generalization of the Concept of a Ring of Quotients

1971 ◽  
Vol 14 (4) ◽  
pp. 517-529 ◽  
Author(s):  
John K. Luedeman
Keyword(s):  
Left Ideal ◽  
Quotient Ring ◽  
Original Method ◽  
Duke Math ◽  
Injective Hull ◽  
Direct Products ◽  
Left Module ◽  
Matrix Rings ◽  
Ring Of Quotients ◽  
Quotient Rings

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

scholarly journals Unique factorization in the ring R[x]

1992 ◽  
Vol 53 (3) ◽  
pp. 287-293 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Left Ideal ◽  
Polynomial Ring ◽  
The Other ◽  
Unique Factorization ◽  
Ideal Domain ◽  
Unique Factorization Domain ◽  
Other Hand ◽  
Reasonable Sense

AbstractIf R is a commutative unique factorization domain (UFD) then so is the ring R[x]. If R is not commutative then no such result is possible. An example is given of a bounded principal right and left ideal domain R, hence a similarity-UFD, for which the polynomial ring R[x] in a central indeterminate x is not a UFD in any reasonable sense. On the other hand, it is shown that if R is an invariant UFD then R[x] is a UFD in an appropriate sense.

Elementary matrix reduction over Bézout domains

2019 ◽  
Vol 18 (08) ◽  
pp. 1950141
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi
Keyword(s):  
Elementary Divisor ◽  
Bezout Domain ◽  
Elementary Matrix ◽  
Matrix Reduction ◽  
Elementary Divisor Ring ◽  
Bezout Domains ◽  
Diagonal Reduction

A ring [Formula: see text] is an elementary divisor ring if every matrix over [Formula: see text] admits a diagonal reduction. If [Formula: see text] is an elementary divisor domain, we prove that [Formula: see text] is a Bézout duo-domain if and only if for any [Formula: see text], [Formula: see text] such that [Formula: see text]. We explore certain stable-like conditions on a Bézout domain under which it is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

Classical Quotient Rings and Ordinary Extensions of 2-Primal Rings

2006 ◽  
Vol 13 (03) ◽  
pp. 513-523 ◽  
Author(s):  
Yong Uk Cho ◽  
Nam Kyun Kim ◽  
Mi Hyang Kwon ◽  
Yang Lee
Keyword(s):  
Local Ring ◽  
Division Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Quotient Rings

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

DECIDABILITY OF MODULES OVER A BÉZOUT DOMAIN D+XQ[X] WITH D A PRINCIPAL IDEAL DOMAIN AND Q ITS FIELD OF FRACTIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 296-305 ◽  
Author(s):  
GENA PUNINSKI ◽  
CARLO TOFFALORI
Keyword(s):  
Principal Ideal ◽  
Principal Ideal Domain ◽  
Bezout Domain ◽  
Ideal Domain ◽  
Ziegler Spectrum

Abstract We describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

On λ-extensions of commutative rings

2018 ◽  
Vol 17 (04) ◽  
pp. 1850063 ◽  
Author(s):  
Rahul Kumar ◽  
Atul Gaur
Keyword(s):  
Automorphism Group ◽  
Local Ring ◽  
Commutative Rings ◽  
Valuation Domain ◽  
Closed Formula ◽  
Integrally Closed

Let [Formula: see text] be commutative rings with identity such that [Formula: see text]. We recall that [Formula: see text] is called a [Formula: see text]-extension of rings if the set of all subrings of [Formula: see text] containing [Formula: see text] (the “intermediate rings”) is linearly ordered under inclusion. In this paper, a characterization of integrally closed [Formula: see text]-extension of rings is given. For example, we show that if [Formula: see text] is a local ring, then [Formula: see text] is an integrally closed [Formula: see text]-extension of rings if and only if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] is a valuation domain. Let [Formula: see text] be a subring of [Formula: see text] such that [Formula: see text] is invariant under action by [Formula: see text], where [Formula: see text] is a subgroup of the automorphism group of [Formula: see text]. If [Formula: see text] is a [Formula: see text]-extension of rings, then [Formula: see text] is a [Formula: see text]-extension of rings under some conditions.

Semiprime Rings with Hypercentral Derivations

1995 ◽  
Vol 38 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Left Ideal ◽  
Semiprime Ring ◽  
Quotient Ring ◽  
Central Idempotent ◽  
Utumi Quotient Ring ◽  
Semiprime Rings ◽  
Positive Integers

AbstractLetRbe a semiprime ring with a derivationd, λ a left ideal ofRandk, ntwo positive integers. Suppose that[d(xn),xn]k= 0 for allx∊ λ. Then [λ,R]d(R)= 0. That is, there exists a central idempotente∊U, the left Utumi quotient ring ofR, such thatdvanishes identically oneUand λ(l —e) is central in (1 —e)U

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