On a Sheaf of Division Rings*

1975 ◽  
Vol 17 (5) ◽  
pp. 727-731
Author(s):  
George Szeto
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Central Idempotent ◽  
Division Rings ◽  
Maximal Ideals ◽  
Strongly Regular ◽  
Th 1

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

WPZI RINGS AND STRONG REGULARITY

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Junchao Wei
Keyword(s):  
Regular Ring ◽  
Strong Regularity ◽  
Strongly Regular

Abstract In this paper, we study the strong regularity of left SF rings and obtain the following results: Let R be a left SF ring. If R satisfies one of the following conditions, then R is a strongly regular ring: 1) R is a left WPZI ring; 2) R is a right WPZI ring; 3) R is a right weakly semicommutative ring; 4) R is a semicommutative ring; 5) R is a reversible ring.

On PIC(D[α]) For a Principal Ideal Domain D

1989 ◽  
Vol 32 (1) ◽  
pp. 114-116 ◽  
Author(s):  
Robert Gilmer ◽  
William Heinzer
Keyword(s):  
Principal Ideal ◽  
Negative Answer ◽  
Picard Group ◽  
Principal Ideal Domain ◽  
Ring Extension ◽  
Simple Ring ◽  
Ideal Domain ◽  
Maximal Ideals

AbstractLet D be a PID with infinitely many maximal ideals. J. W. Brewer has asked whether some simple ring extension D[α] of D must have nontrivial Picard group. We show that this question has a negative answer.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

2019 ◽  
Vol 26 (02) ◽  
pp. 181-194 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Principal Ideal ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Chain Ring ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Ideal Ring ◽  
Maximal Ideals

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

scholarly journals Positive derivations on partially ordered strongly regular rings

1984 ◽  
Vol 37 (2) ◽  
pp. 178-180 ◽  
Author(s):  
D. J. Hansen
Keyword(s):  
Regular Ring ◽  
Additional Property ◽  
Partially Ordered ◽  
Strongly Regular ◽  
Regular Rings

AbstractThe author presents a proof that a partially ordered strongly regular ring S which has the additional property that the square of each member of S is greater than or equal to zero cannot have nontrivial positive derivations.

A note on modules over regular rings

1971 ◽  
Vol 4 (1) ◽  
pp. 57-62 ◽  
Author(s):  
K. M. Rangaswamy ◽  
N. Vanaja
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Free Left ◽  
Von Neumann Regular ◽  
Strongly Regular ◽  
Torsion Free ◽  
Regular Rings

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

Rings whose cyclics are C3-modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650152 ◽  
Author(s):  
Yasser Ibrahim ◽  
Xuan Hau Nguyen ◽  
Mohamed F. Yousif ◽  
Yiqiang Zhou
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Matrix Ring ◽  
Artinian Ring ◽  
Local Rings ◽  
Classical Theorem ◽  
Basic Properties ◽  
Semiperfect Ring ◽  
Injectivity Condition ◽  
Strongly Regular

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

Some model theory of modules. II. on stability and categoricity of flat modules

1983 ◽  
Vol 48 (4) ◽  
pp. 970-985 ◽  
Author(s):  
Philipp Rothmaler
Keyword(s):  
Model Theory ◽  
Regular Ring ◽  
Main Tool ◽  
Categorical Theory ◽  
Maximal Ideals ◽  
Flat Modules ◽  
Semisimple Ring ◽  
Definable Sets ◽  
The Stability ◽  
Regular Rings

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

scholarly journals Rings with Central Idempotent or Nilpotent Elements

1958 ◽  
Vol 9 (4) ◽  
pp. 157-165 ◽  
Author(s):  
M. P. Drazin
Keyword(s):  
Regular Ring ◽  
Associative Ring ◽  
Idempotent Element ◽  
Central Idempotent ◽  
Nilpotent Elements ◽  
Regular Rings

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

Morphic Properties of Extensions of Rings

2010 ◽  
Vol 17 (02) ◽  
pp. 337-344
Author(s):  
Qinghe Huang ◽  
Jianlong Chen
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Group Ring ◽  
Regular Ring ◽  
Strongly Regular ◽  
Left Annihilator

An element a in a ring R is called left morphic if R/Ra ≅ ℓR(a), where ℓR(a) denotes the left annihilator of a in R. A ring R is said to be left morphic if every element is left morphic. In this paper, it is shown that if I is an ideal of a unit regular ring R, then for each positive integer n, [Formula: see text] is a left morphic ring. This extends two recent results of Lee and Zhou. It is also proved that if R is a strongly regular ring and Cn= 〈g〉 is a cyclic group of order n ≥ 2, then for any r ∈ R, 1 + rg is morphic in the group ring RCn.

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