On 
	    
	    P
	    
	  Wπ-regular rings

Abstract As a popularization of weakly π-regular rings, we tender the connotation of W P Wπ-regular rings, that is if for each 𝔞 ∈ Ɉ(𝔑), there exist a natural number 𝔫 such that 𝔞𝔫 ∈ 𝔞𝔫 𝔑 𝔞𝔫 𝔑 (𝔞𝔫 ∈ 𝔑 𝔞𝔫 𝔑 𝔞𝔫). In this treatise, numerous properties of this sort of rings are discussed, some important results are secured. Using the connotation of P Wπ-regular rings. It is show that : 1- Let 𝔑 be a right P Wπ-regular ring and ℵɈ-rings with 𝔞𝔫𝔑 = 𝔑𝔞𝔫 for every 𝔞 ∈ Ɉ(𝔑) and for at least one of a natural number. Then Ɉ(𝔑) = ℵ𝔑. 2- Let 𝔑 a right P Wπ-regular ring and 𝔞𝔑 = 𝔑𝔞 for each ∈Ɉ(𝔑). Then 𝔑 is right P .𝔗-ring. 3 Let 𝔑 be a ring with ɍ(𝔞) ⊆ ɭ(𝔞), for each ∈ Ɉ (𝔑). If any of the next conditions are hold, then 𝔑 is P Wπ-regular rings : i – Every maximal right ideal of 𝔑 is a right annihilator and right Ɉ PP -ring. ii – any simple singular right 𝔑-module is Ɉ-injective and 𝔑 is semi prime.

Commutative von Neumann regular rings are 1-Gröbner

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm210419030p ◽

2021 ◽

pp. 30-30

Author(s):

Zoran Petrovic ◽

Maja Roslavcev

Keyword(s):

Regular Ring ◽

Finitely Generated ◽

Von Neumann ◽

Von Neumann Regular ◽

Ring Of Polynomials ◽

Bner Basis ◽

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.

Factor-Correspondences in Regular Rings

10.4153/cjm-1982-004-0 ◽

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 ◽

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

10.4153/cjm-1986-032-0 ◽

1986 ◽

Vol 38 (3) ◽

pp. 633-658 ◽

Cited By ~ 17

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 ◽

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].

Centres of Rank-Metric Completions

10.4153/cjm-1985-061-7 ◽

1985 ◽

Vol 37 (6) ◽

pp. 1134-1148

Author(s):

David Handelman

Keyword(s):

Regular Ring ◽

Rank Function ◽

Von Neumann ◽

Rank Metric ◽

Completion Process ◽

Von Neumann Regular ◽

Metric Topology ◽

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-055-1 ◽

1974 ◽

Vol 17 (2) ◽

pp. 283-284 ◽

Cited By ~ 1

Author(s):

Kwangil Koh

Keyword(s):

Regular Ring ◽

The Other ◽

Von Neumann ◽

Research Problems ◽

Von Neumann Regular ◽

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.

Radical Regularity in Differential Rings

10.4153/cjm-1971-019-7 ◽

1971 ◽

Vol 23 (2) ◽

pp. 197-201 ◽

Cited By ~ 6

Author(s):

Howard E. Gorman

Keyword(s):

Jacobson Radical ◽

Regular Ring ◽

Prime Ideals ◽

Differential Ring ◽

Quotient Maps ◽

The Stability ◽

Definition Of ◽

Finitely Generated Modules ◽

Regular Rings ◽

The Ideal

In [1], we discussed completions of differentially finitely generated modules over a differential ring R. It was necessary that the topology of the module be induced by a differential ideal of R and it was natural that this ideal be contained in J(R), the Jacobson radical of R. The ideal to be chosen, called Jd(R), was the intersection of those ideals which are maximal among the differential ideals of R. The question as to when Jd(R) ⊆ J(R) led to the definition of a class of rings called radically regular rings. These rings do satisfy the inclusion, and we showed in [1, Theorem 2] that R could always be “extended”, via localization, to a radically regular ring in such a way as to preserve all its differential prime ideals.In the present paper, we discuss the stability of radical regularity under quotient maps, localization, adjunction of a differential indeterminate, and integral extensions.

A Theorem About Maximal Cohen–Macaulay Modules

International Mathematics Research Notices ◽

10.1093/imrn/rnaa154 ◽

2020 ◽

Author(s):

Thomas Polstra

Keyword(s):

Natural Number ◽

Regular Ring ◽

Class Group ◽

Torsion Subgroup ◽

Finitely Generated ◽

Divisor Class Group ◽

Divisor Class ◽

Frobenius Endomorphism

Abstract It is shown that for any local strongly $F$-regular ring there exists natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen–Macaulay module, then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.

TWO QUESTIONS OF L. VAŠ ON -CLEAN RINGS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713000117 ◽

2013 ◽

Vol 88 (3) ◽

pp. 499-505 ◽

Cited By ~ 3

Author(s):

JIANLONG CHEN ◽

JIAN CUI

Keyword(s):

Regular Ring ◽

Von Neumann Algebras ◽

Von Neumann ◽

Clean Rings ◽

AbstractA $\ast $-ring $R$ is called (strongly) $\ast $-clean if every element of $R$ is the sum of a unit and a projection (that commute). Vaš [‘$\ast $-Clean rings; some clean and almost clean Baer $\ast $-rings and von Neumann algebras’, J. Algebra 324(12) (2010), 3388–3400] asked whether there exists a $\ast $-ring that is clean but not $\ast $-clean and whether a unit regular and $\ast $-regular ring is strongly $\ast $-clean. In this paper, we answer these two questions. We also give some characterisations related to $\ast $-regular rings.

Invo-regular unital rings

Annales Universitatis Mariae Curie-Sklodowska sectio A – Mathematica ◽

10.17951/a.2018.72.1.45-53 ◽

2018 ◽

Vol 72 (1) ◽

pp. 45

Author(s):

Peter V. Danchev

Keyword(s):

Regular Ring ◽

A Priori ◽

Proper Subclass ◽

In Trans ◽

Unital Rings ◽

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

Onn-flat modules andn-Von Neumann regular rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/90868 ◽

2006 ◽

Vol 2006 ◽

pp. 1-6

Author(s):

Najib Mahdou

Keyword(s):

Regular Ring ◽

Local Rings ◽

Von Neumann ◽

Flat Modules ◽

Von Neumann Regular ◽

We show that eachR-module isn-flat (resp., weaklyn-flat) if and only ifRis an(n,n−1)-ring (resp., a weakly(n,n−1)-ring). We also give a new characterization ofn-Von Neumann regular rings and a characterization of weakn-Von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weakn-Von Neumann regular ring is a (CH)-ring.