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).

scholarly journals On (von Neumann) regular rings

1974 ◽  
Vol 19 (1) ◽  
pp. 89-91 ◽  
Author(s):  
R. Yue Chi Ming
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Associative Ring ◽  
Simple Module ◽  
Short Note ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Left Annihilator ◽  
Regular Rings

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

scholarly journals On strongly pi-regular rings of stable range one

1995 ◽  
Vol 51 (3) ◽  
pp. 433-437 ◽  
Author(s):  
Hua-Ping Yu ◽  
Victor P. Camilo
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Associative Ring ◽  
Stable Range ◽  
Von Neumann ◽  
Stable Range One ◽  
Von Neumann Regular ◽  
Open Question ◽  
Regular Rings

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

scholarly journals Commutative von Neumann regular rings are 1-Gröbner

2021 ◽  
pp. 30-30
Author(s):  
Zoran Petrovic ◽  
Maja Roslavcev
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Polynomials ◽  
Bner Basis ◽  
Regular Rings

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

2011 ◽  
Vol 21 (05) ◽  
pp. 745-762 ◽  
Author(s):  
TAI KEUN KWAK ◽  
YANG LEE
Keyword(s):  
Polynomial Ring ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

Factor-Correspondences in Regular Rings

1982 ◽  
Vol 34 (1) ◽  
pp. 23-30
Author(s):  
S. K. Berberian
Keyword(s):  
Present Article ◽  
Regular Ring ◽  
Decomposition Theorem ◽  
Matrix Rings ◽  
Basic Properties ◽  
Principal Ideals ◽  
Highly Effective ◽  
Left Factor ◽  
Regular Rings

Factor-correspondences are nothing more than a way of describing isomorphisms between principal ideals in a regular ring. However, due to a remarkable decomposition theorem of M. J. Wonenburger [7, Lemma 1], they have proved to be a highly effective tool in the study of completeness properties in matrix rings over regular rings [7, Theorem 1]. Factor-correspondences also figure in the proof of D. Handelman's theorem that an ℵ0-continuous regular ring is unitregular [4, Theorem 3.2].The aim of the present article is to sharpen the main result in [7] and to re-examine its applications to matrix rings. The basic properties of factor-correspondences are reviewed briefly for the reader's convenience.Throughout, R denotes a regular ring (with unity).Definition 1 (cf. [5, p. 209ff], [7, p. 212]). A right factor-correspondence in R is a right R-isomorphism φ : J → K, where J and K are principal right ideals of R (left factor-correspondences are defined dually).

Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

1986 ◽  
Vol 38 (3) ◽  
pp. 633-658 ◽  
Author(s):  
K. R. Goodearl ◽  
D. E. Handelman
Keyword(s):  
Regular Ring ◽  
Hausdorff Space ◽  
Tensor Products ◽  
Matrix Algebras ◽  
Dimension Group ◽  
Grothendieck Groups ◽  
Dimension Groups ◽  
Direct Limits ◽  
Finite Products ◽  
Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

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.

Centres of Rank-Metric Completions

1985 ◽  
Vol 37 (6) ◽  
pp. 1134-1148
Author(s):  
David Handelman
Keyword(s):  
Regular Ring ◽  
Rank Function ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Rank Metric ◽  
Von Neumann Regular Rings ◽  
Completion Process ◽  
Von Neumann Regular ◽  
Metric Topology ◽  
Regular Rings

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

On Von Neumann Regular Rings

1974 ◽  
Vol 17 (2) ◽  
pp. 283-284 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Regular Ring ◽  
The Other ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Research Problems ◽  
Von Neumann Regular ◽  
Regular Rings

Recently, in the Research Problems of Canadian Mathematical Bulletin, Vol. 14, No. 4, 1971, there appeared a problem which asks “Is a prime Von Neumann regular ring pimitive?” While we are not able to settle this question one way or the other, we prove that in a Von Neumann regular ring, there is a maximal annihilator right ideal if and only if there is a minimal right ideal.

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