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.

Semisimplicity of Free Centred Extensions

1995 ◽  
Vol 38 (1) ◽  
pp. 55-58 ◽  
Author(s):  
Miguel Ferrero
Keyword(s):  
Extended Centroid ◽  
Semisimple Ring

AbstractWe prove that a free centred extension R[E] is a semisimple ring if R is a semisimple ring and C[E] is semisimple for every field C which is the extended centroid of a primitive factor of R.

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

Ideals of topological ring S?

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

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]).

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.

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