Ideals of topological ring S?

1973 ◽  
Vol 7 (1) ◽  
pp. 78-79 ◽  
Author(s):  
V. � Semenova
Keyword(s):  
Topological Ring

Dual-Parameter Fiber-Optics Relative Humidity and Displacement Sensor Based on the Topological Ring Structures

2021 ◽  
pp. 1-1
Author(s):  
Ying Guo ◽  
Yundong Zhang ◽  
Huaiyin Su ◽  
Kaiyue Qi ◽  
Fuxing Zhu ◽  
...  
Keyword(s):  
Relative Humidity ◽  
Fiber Optics ◽  
Displacement Sensor ◽  
Topological Ring ◽  
Ring Structures

On the Set of Zero Divisors of a Topological Ring

1967 ◽  
Vol 10 (4) ◽  
pp. 595-596 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Finite Number ◽  
Compact Subset ◽  
Finite Ring ◽  
Left Zero ◽  
Topological Ring ◽  
Compact Ring ◽  
Zero Divisors ◽  
Interesting Corollary

Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra are closed subsets of R. We will prove that if the set of non - trivial right (left) zero divisors of R is a non-empty set and the set of all right (left) zero divisors of R is a compact subset of R, then R is a compact ring. This theorem has an interesting corollary. Namely, if R is a discrete ring with a finite number of non - trivial left or right zero divisors then R is a finite ring (Refer [1]).

Ideals in Topological Rings

1964 ◽  
Vol 16 ◽  
pp. 28-45 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Topological Ring ◽  
Semisimple Ring

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

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].

On Rings of Fractions

1970 ◽  
Vol 13 (4) ◽  
pp. 425-430 ◽  
Author(s):  
T. M. K. Davison
Keyword(s):  
Noetherian Ring ◽  
Multiplicative System ◽  
Topological Ring ◽  
Commutative Noetherian Ring ◽  
Image Position ◽  
Fixed Ideal

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R. Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R, and let j = js: R → S-1R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S-1R is a topological ring in ihe j(M)-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p(S), whereHence S-1R is a topological ring for all S if and only if M ⊂ p*(R), where

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

The Complete Quotient Ring of Images of Semilocal Prüfer Domains

1977 ◽  
Vol 29 (5) ◽  
pp. 914-927 ◽  
Author(s):  
John Chuchel ◽  
Norman Eggert
Keyword(s):  
Homomorphic Image ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Local Rings ◽  
Topological Ring ◽  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Quotient Rings ◽  
Prüfer Rings

It is well known that the complete quotient ring of a Noetherian ring coincides with its classical quotient ring, as shown in Akiba [1]. But in general, the structure of the complete quotient ring of a given ring is largely unknown. This paper investigates the structure of the complete quotient ring of certain Prüfer rings. Boisen and Larsen [2] considered conditions under which a Prüfer ring is a homomorphic image of a Prüfer domain and the properties inherited from the domain. We restrict our investigation primarily to homomorphic images of semilocal Prüfer domains. We characterize the complete quotient ring of a semilocal Prüfer domain in terms of complete quotient rings of local rings and a completion of a topological ring.

scholarly journals On the Structure of Complete Local Rings

1950 ◽  
Vol 1 ◽  
pp. 63-70 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Structure Theorem ◽  
Finite Basis ◽  
Local Rings ◽  
Topological Ring

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

scholarly journals Решеточная структура в пространстве ограниченных гомоморфизмов между топологическими решеточно упорядоченными кольцами

2019 ◽  
Author(s):  
O. Zabeti
Keyword(s):  
Lattice Structure ◽  
Bounded Operators ◽  
Topological Ring ◽  
Topological Structures ◽  
Riesz Spaces ◽  
Topological Lattice ◽  
Bounded Group

Suppose X is a topological ring. It is known that there are three classes of bounded group homomorphisms on X whose topological structures make them again topological rings. First, we show that if X is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on X. Now, assume that X is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on X. Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable RieszKantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups.

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