Rings All of whose Factor Rings are Semi-Prime

1969 ◽  
Vol 12 (4) ◽  
pp. 417-426 ◽  
Author(s):  
R.C. Courter
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Title One ◽  
Regular Rings

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.

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.

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 The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

2020 ◽  
Author(s):  
Pere Ara ◽  
Joan Bosa ◽  
Enrique Pardo ◽  
Aidan Sims
Keyword(s):  
Inverse Semigroup ◽  
Finitely Generated ◽  
Realization Problem ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Refinement Monoids ◽  
Étale Groupoid ◽  
Regular Rings

Abstract Given an adaptable separated graph, we construct an associated groupoid and explore its type semigroup. Specifically, we first attach to each adaptable separated graph a corresponding semigroup, which we prove is an $E^*$-unitary inverse semigroup. As a consequence, the tight groupoid of this semigroup is a Hausdorff étale groupoid. We show that this groupoid is always amenable and that the type semigroups of groupoids obtained from adaptable separated graphs in this way include all finitely generated conical refinement monoids. The first three named authors will utilize this construction in forthcoming work to solve the realization problem for von Neumann regular rings, in the finitely generated case.

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.

scholarly journals Onn-flat modules andn-Von Neumann regular rings

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Najib Mahdou
Keyword(s):  
Regular Ring ◽  
Local Rings ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Flat Modules ◽  
Von Neumann Regular ◽  
Regular Rings

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.

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.

scholarly journals On strongly coseparable modules

2021 ◽  
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Free Module ◽  
Finitely Generated ◽  
Direct Sums ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

AbstractA module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

Regular Rings are Very Regular

1982 ◽  
Vol 25 (1) ◽  
pp. 118-118 ◽  
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Finitely Generated ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Regular Rings

The following problem arose in a conversation with Abraham Zaks: “Suppose R is an associative ring with identity such that every finitely generated left ideal is generated by idempotents. Is R von-Neumann regular?” In the literature the “s” in “idempotents” is missing, and is replaced by “an idempotent”. The answer is, “Yes!”

On (A)-rings and strong (A)-rings issued from amalgamations

2018 ◽  
Vol 55 (2) ◽  
pp. 270-279 ◽  
Author(s):  
Najib Mahdou ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Ring Homomorphism ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Von Neumann ◽  
Zero Divisors ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Regular Rings

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.

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