Topological Rings of Quotients

1974 ◽  
Vol 26 (5) ◽  
pp. 1228-1233 ◽  
Author(s):  
William Schelter
Keyword(s):  
Compact Space ◽  
Topological Ring ◽  
Ring Of Quotients ◽  
Algebraic Ring ◽  
Rings Of Quotients

We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

Clifford semigroups of left quotients

1986 ◽  
Vol 28 (2) ◽  
pp. 181-191 ◽  
Author(s):  
Victoria Gould
Keyword(s):  
Ring Theory ◽  
Clifford Semigroups ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Classical Ring ◽  
Number Of Authors

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].

scholarly journals Filters and overrings

1975 ◽  
Vol 19 (4) ◽  
pp. 474-480
Author(s):  
H. H. Brungs
Keyword(s):  
Integral Domain ◽  
Ring Of Quotients ◽  
One To One ◽  
Rings Of Quotients

Let R be an integral domain. It is well known (see Lambek (1971), Stenström (1971)), that idempotent filters of right ideals, torsion radicals and trosio theories are in one-to-one correspondence, but that different idempotent filters F of right ideals may lead to the same rings of quotiens Rf. We have always R ⊃ Rf ⊂Qmax(R). Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Qmax(R) of the form RF. When is every RF of the form RS−1, where S is an Ore system?

III.—Multiplicative Ideal Theory and Rings of Quotients. I.

1968 ◽  
Vol 68 (1) ◽  
pp. 30-53
Author(s):  
George D. Findlay
Keyword(s):  
Associative Ring ◽  
Ideal Theory ◽  
Maximal Order ◽  
Ring Of Quotients ◽  
Suitable Generalization ◽  
Rings Of Quotients

SynopsisBy means of a generalized ring of quotients multiplicative ideal theory is studied in an arbitrary (associative) ring. A suitable generalization of the concept of maximal order is given and factorization theorems are obtained for the nonsingular (two sided) ideals, which generalize the theorems of Artin and E. Noether.

scholarly journals Semisimple rings of quotients

1978 ◽  
Vol 19 (1) ◽  
pp. 97-115 ◽  
Author(s):  
Julius M. Zelmanowitz
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Necessary And Sufficient

Necessary and sufficient conditions on an arbitrary Gabriel filter of left ideals of a ring R are determined in order that the ring of quotients of R with respect to the filter be semi-simple artinian. Special instances include generalizations of earlier work on classical rings of quotients and maximal rings of quotients.

scholarly journals Covering classes and 1-tilting cotorsion pairs over commutative rings

2021 ◽  
Vol 33 (3) ◽  
pp. 601-629
Author(s):  
Silvana Bazzoni ◽  
Giovanna Le Gros
Keyword(s):  
New York ◽  
Projective Dimension ◽  
Commutative Rings ◽  
Cotorsion Pair ◽  
Bijective Correspondence ◽  
Ring Of Quotients ◽  
Cotorsion Pairs ◽  
Rings Of Quotients ◽  
Covering Class ◽  
Quotient Rings

Abstract We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜 {\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 {\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} , and R 𝒢 {R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢 {\mathcal{G}} , we show that if 𝒜 {\mathcal{A}} is covering, then 𝒢 {\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R 𝒢 {R_{\mathcal{G}}} has projective dimension at most one as an R-module. Moreover, we show that 𝒜 {\mathcal{A}} is covering if and only if both the localisation R 𝒢 {R_{\mathcal{G}}} and the quotient rings R / J {R/J} are perfect rings for every J ∈ 𝒢 {J\in\mathcal{G}} . Rings satisfying the latter two conditions are called 𝒢 {\mathcal{G}} -almost perfect.

Lattice-Ordered Rings of Quotients

1965 ◽  
Vol 17 ◽  
pp. 434-448 ◽  
Author(s):  
F. W. Anderson
Keyword(s):  
Division Ring ◽  
Integral Domain ◽  
Special Cases ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Ore Domain

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.

Rings of Quotients of Rings of Derivations

1968 ◽  
Vol 11 (3) ◽  
pp. 383-398
Author(s):  
Israel Kleiner
Keyword(s):  
Ring Structure ◽  
Lie Ring ◽  
Lie Rings ◽  
Rational Extension ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Associative Rings

The concept of a rational extension of a Lie module is defined as in the associative case [1, pp. 81 and 79]. It then follows from [3, Theorem 2.3] that any Lie module possesses a maximal rational extension (a rational completion), unique up to isomorphism. If now L and K are Lie rings with L⊆ K, we call K a (Lie) ring of quotients of L if K, considered as a Lie module over L, is a rational extension of the Lie module LL. Although we do not know if for every Lie ring L its rational completion can be given a Lie ring structure extending that of L (as is the case for associative rings), this is so, in any case, for abelian Lie rings (Propositions 2 and 4).

scholarly journals PERFECT SYMMETRIC RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 689-711 ◽  
Author(s):  
LIA VAŠ
Keyword(s):  
The Other ◽  
Commutative Case ◽  
Perfect Ring ◽  
Other Hand ◽  
Standard Construction ◽  
Desirable Feature ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
The Right

Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter — the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients [Formula: see text] can be adapted to the construction of the largest perfect symmetric ring of quotients — the total symmetric ring of quotients [Formula: see text]. We also demonstrate that Morita's construction of [Formula: see text] can be adapted to the construction of [Formula: see text].

Continuous Rings and Rings of Quotients

1978 ◽  
Vol 21 (3) ◽  
pp. 319-324 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Jacobson Radical ◽  
Associative Ring ◽  
Natural Generalization ◽  
Injective Hull ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Rings Of Quotients

Throughout R will denote an associative ring with identity. Let Zℓ(R) be the left singular ideal of R. It is well known that Zℓ(R) = 0 if and only if the left maximal ring of quotients of R, Q(R), is Von Neumann regular. When Zℓ(R) = 0, q(R) is also a left self injective ring and is, in fact, the injective hull of R. A natural generalization of the notion of injective is the concept of left continuous as studied by Utumi [4]. One of the major obstacles to studying the relationships between Q(R) and R is a description of J(Q(R)), the Jacobson radical of Q(R). When a ring is left continuous, then its left singular ideal is its Jacobson radical. This facilitates the study of the cases when either Q(R) is continuous or R is continuous.

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