scholarly journals Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

2003 ◽  
Vol 180 (3) ◽  
pp. 221-241 ◽  
Author(s):  
David F. Anderson ◽  
Ron Levy ◽  
Jay Shapiro
Keyword(s):  
Zero Divisor ◽  
Boolean Algebras ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

scholarly journals On zero-dimensionality and fragmented rings

1999 ◽  
Vol 60 (1) ◽  
pp. 137-151
Author(s):  
Jim Coykendall ◽  
David E. Dobbs ◽  
Bernadette Mullins
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Krull Dimension ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Finite Products ◽  
Regular Rings

A commutative ring R is said to be fragmented if each nonunit of R is divisible by all positive integral powers of some corresponding nonunit of R. It is shown that each fragmented ring which contains a nonunit non-zero-divisor has (Krull) dimension ∞. We consider the interplay between fragmented rings and both the atomic and the antimatter rings. After developing some results concerning idempotents and nilpotents in fragmented rings, along with some relevant examples, we use the “fragmented” and “locally fragmented” concepts to obtain new characterisations of zero-dimensional rings, von Neumann regular rings, finite products of fields, and fields.

Spectrum of the zero-divisor graph of von Neumann regular rings

2021 ◽  
pp. 2250193
Author(s):  
Avinash Patil ◽  
Kiran Shinde
Keyword(s):  
Zero Divisor ◽  
Laplacian Matrix ◽  
Induced Subgraphs ◽  
Von Neumann ◽  
Laplacian Eigenvalues ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

The zero-divisor graph [Formula: see text] of a commutative ring [Formula: see text] is the graph whose vertices are the nonzero zero divisors in [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph [Formula: see text] of a finite commutative von Neumann regular ring [Formula: see text]. We prove that [Formula: see text] is a generalized join of its induced subgraphs. Among the [Formula: see text] eigenvalues (respectively, Laplacian eigenvalues) of [Formula: see text], exactly [Formula: see text] are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of [Formula: see text]-the zero-divisor graph of nontrivial idempotents in [Formula: see text]. We also determine the degree of each vertex in [Formula: see text], hence the number of edges.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

scholarly journals Normal Subgroups of Classical Groups over von Neumann Regular Rings

1994 ◽  
Vol 169 (3) ◽  
pp. 863-873
Author(s):  
F.A. Arlinghaus ◽  
L.N. Vaserstein ◽  
H. You
Keyword(s):  
Classical Groups ◽  
Normal Subgroups ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

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.

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