Commutative von Neumann regular rings are 1-Gröbner

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.

Centres of Rank-Metric Completions

Canadian Journal of Mathematics ◽

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.

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.

A note on modules over regular rings

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046268 ◽

1971 ◽

Vol 4 (1) ◽

pp. 57-62 ◽

Cited By ~ 3

Author(s):

K. M. Rangaswamy ◽

N. Vanaja

Keyword(s):

Regular Ring ◽

Finitely Generated ◽

Von Neumann ◽

Free Left ◽

Von Neumann Regular ◽

Strongly Regular ◽

Torsion Free ◽

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 All of whose Factor Rings are Semi-Prime

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-052-2 ◽

1969 ◽

Vol 12 (4) ◽

pp. 417-426 ◽

Cited By ~ 16

Author(s):

R.C. Courter

Keyword(s):

Left Ideal ◽

Regular Ring ◽

Finitely Generated ◽

Von Neumann ◽

Von Neumann Regular ◽

Title One ◽

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.

On the torsion graph and von Neumann regular rings

Filomat ◽

10.2298/fil1202253r ◽

2012 ◽

Vol 26 (2) ◽

pp. 253-259

Author(s):

Malakooti Rad ◽

S.H. Ghalandarzadeh ◽

S. Shirinkam

Keyword(s):

Complete Graph ◽

Regular Ring ◽

Von Neumann ◽

Prime Submodules ◽

Von Neumann Regular ◽

Minimal Prime ◽

The Relationship ◽

Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by ?(M), is a graph whose vertices are the non-zero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = 0. In this paper, we investigate the relationship between the diameters of ?(M) and ?(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a B?zout ring R the diameter of ?(M) and ?(R) is equal, where M , T(M). Also, we prove that, for a faithful multiplication R-module M with |M|?4,?(M) is a complete graph if and only if ?(R) is a complete graph.

The Groupoids of Adaptable Separated Graphs and Their Type Semigroups

International Mathematics Research Notices ◽

10.1093/imrn/rnaa022 ◽

2020 ◽

Cited By ~ 1

Author(s):

Pere Ara ◽

Joan Bosa ◽

Enrique Pardo ◽

Aidan Sims

Keyword(s):

Inverse Semigroup ◽

Finitely Generated ◽

Realization Problem ◽

Von Neumann ◽

Von Neumann Regular ◽

Refinement Monoids ◽

Étale Groupoid ◽

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.

J-Regular Rings with Injectivities

Algebra Colloquium ◽

10.1142/s100538671300031x ◽

2013 ◽

Vol 20 (02) ◽

pp. 343-347 ◽

Cited By ~ 1

Author(s):

Liang Shen

Keyword(s):

Jacobson Radical ◽

Regular Ring ◽

Von Neumann ◽

Von Neumann Regular ◽

Let R be a J-regular ring, i.e., R/J(R) is a von Neumann regular ring, where J(R) is the Jacobson radical of R. It is proved: (i) For every n ≥ 1, R is right n-injective if and only if every homomorphism from an n-generated small right ideal of R to RR can be extended to one from RR to RR. (ii) R is right FP-injective if and only if R is right (J,R)-FP-injective. Some known results are improved.

On strongly coseparable modules

Arabian Journal of Mathematics ◽

10.1007/s40065-021-00333-1 ◽

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 ◽

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.

On (von Neumann) regular rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500015418 ◽

1974 ◽

Vol 19 (1) ◽

pp. 89-91 ◽

Cited By ~ 17

Author(s):

R. Yue Chi Ming

Keyword(s):

Left Ideal ◽

Regular Ring ◽

Associative Ring ◽

Simple Module ◽

Short Note ◽

Von Neumann ◽

Von Neumann Regular ◽

Left Annihilator ◽

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.