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

Von Neumann Regular ◽

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.

THE ZERO-DIVISOR GRAPH OF VON NEUMANN REGULAR RINGS

Communications in Algebra ◽

10.1081/agb-120013178 ◽

2002 ◽

Vol 30 (2) ◽

pp. 745-750 ◽

Cited By ~ 48

Author(s):

Ron Levy ◽

Jay Shapiro

Keyword(s):

Zero Divisor ◽

Von Neumann ◽

Zero Divisor Graph ◽

Von Neumann Regular ◽

On zero-dimensionality and fragmented rings

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700033402 ◽

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 ◽

Finite Products ◽

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.

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

Journal of Pure and Applied Algebra ◽

10.1016/s0022-4049(02)00250-5 ◽

2003 ◽

Vol 180 (3) ◽

pp. 221-241 ◽

Cited By ~ 97

Author(s):

David F. Anderson ◽

Ron Levy ◽

Jay Shapiro

Keyword(s):

Zero Divisor ◽

Boolean Algebras ◽

Von Neumann ◽

Von Neumann Regular ◽

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

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2018.55.2.1397 ◽

2018 ◽

Vol 55 (2) ◽

pp. 270-279 ◽

Cited By ~ 1

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 ◽

Ring Of Quotients ◽

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.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

Journal of Algebra and Its Applications ◽

10.1142/s0219498809003527 ◽

2009 ◽

Vol 08 (05) ◽

pp. 601-615

Author(s):

JOHN D. LAGRANGE

Keyword(s):

Commutative Rings ◽

Von Neumann ◽

Von Neumann Regular ◽

Ring Of Quotients ◽

Rings Of Quotients ◽

Special Case ◽

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.