CYCLES AND SYMMETRIES OF ZERO-DIVISORS

By Shashikant Mulay. 2002

231 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c625d3c1f1c74686b7e77081a441b73770bcc0e3

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Citations in the corpus

Zero-Divisor Graphs of Matrices Over Commutative Rings (2009, background): “The results from the following theorem were proved in Anderson and Livingston (1999) and Mulay (2002).”
Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings (2005, background): “Communications in Algebra®, 33: 2043–2050, 2005 Copyright © Taylor & Francis, Inc. ISSN: 0092-7872 print/1532-4125 online DOI: 10.1081/AGB-200063357 ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS# Michael Axtell Department of Mathematics and Computer Science, Wabash Colle”
An Ideal-Based Zero-Divisor Graph of a Commutative Ring (2003, background): “The zero-divisor graph of a commutative ring is further examined in Anderson et al. (preprint, 2001), Levy and Shapiro (2002) and Mulay (2002).”
The Annihilating-Ideal Graph of Commutative Rings I (2008, background): “If X is either an element or a subset of R, then the annihilator of X is Ann(X) = {r ∈ R|rX = 0}.”
On the Zero-Divisor Graph of a Ring (2008, background): “If R is an infinite star graph, then either R D× 2 for D an integral domain (when R is reduced), or Nil R is a prime ideal of R with Nil R ∗ = 1 and Z R is a prime ideal of R (DeMeyer and Schneider, 2002, Theorem 1.12 or Mulay, 2002, (2.1)).”
Commutative rings with toroidal zero-divisor graphs (2007, background): “Mulay [15] showed that if Γ(R) contains a cycle, then Γ(R) contains a 3-cycle or a 4-cycle.”
A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors (2007, background): “In particular, one might ask if there exists a ring R such that E R is an infinite star graph, or perhaps if the assumption that R is Noetherian forces E R to be finite.”
On Zero Divisor Graphs (2012, methodology): “13 ([29]).”
Zero-divisor graphs, von Neumann regular rings, and Boolean algebras (2003, background): “The zero-divisor graph of a commutative ring has also been studied in [5,3,9,18,19], and the zero-divisor graph concept has recently been extended to noncommutative rings in [22] and commutative semigroups in [8].”
THE ZERO-DIVISOR GRAPH OF VON NEUMANN REGULAR RINGS (2002, unknown): “In [2, 3, 5] and [7] the authors, using the same vertex set as here, were interested in examining the interplay between the ring-theoretic properties of R and the graph-theoretic properties of GðRÞ.”
Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph (2012, background): “It is observed in [22] that ∼ is an equivalence relation on R.”
Zero-divisor graphs of genus one (2006, background): “Recently, this subject has been studied extensively in [1–8].”
When a zero-divisor graph is planar or a complete r-partite graph (2003, unknown): “(In [9, 1.4] Mulay has shown that this result holds for any commutative ring; also see [6, Theorem 1.6].)”
The Annihilating-Ideal Graph of Commutative Rings II (2008, unknown): “Moreover, if Γ(R) contains a cycle, then gr(Γ(R)) ≤ 4 (see [7]).”
Zero divisor graphs of semigroups (2005, background): “The following result for commutative rings is in [6].”