1,855 research outputs found
Categorial properties of compressed zero-divisor graphs of finite commutative rings
We define a compressed zero-divisor graph of a finite
commutative unital ring , where the compression is performed by means of the
associatedness relation. We prove that this is the best possible compression
which induces a functor , and that this functor preserves categorial
products (in both directions). We use the structure of to
characterize important classes of finite commutative unital rings, such as
local rings and principal ideal rings.Comment: 14 page
Independent sets of some graphs associated to commutative rings
Let be a simple graph. A set is independent set of
, if no two vertices of are adjacent. The independence number
is the size of a maximum independent set in the graph. %An
independent set with cardinality Let be a commutative ring with nonzero
identity and an ideal of . The zero-divisor graph of , denoted by
, is an undirected graph whose vertices are the nonzero
zero-divisors of and two distinct vertices and are adjacent if and
only if . Also the ideal-based zero-divisor graph of , denoted by
, is the graph which vertices are the set {x\in R\backslash I |
xy\in I \quad for some \quad y\in R\backslash I\} and two distinct vertices
and are adjacent if and only if . In this paper we study the
independent sets and the independence number of and .Comment: 27 pages. 22 figure
Rings whose total graphs have genus at most one
Let be a commutative ring with its set of zero-divisors. In this
paper, we study the total graph of , denoted by \T(\Gamma(R)). It is the
(undirected) graph with all elements of as vertices, and for distinct , the vertices and are adjacent if and only if .
We investigate properties of the total graph of and determine all
isomorphism classes of finite commutative rings whose total graph has genus at
most one (i.e., a planar or toroidal graph). In addition, it is shown that,
given a positive integer , there are only finitely many finite rings whose
total graph has genus .Comment: 7 pages. To appear in Rocky Mountain Journal of Mathematic
Properties of Ideal-Based Zero-Divisor Graphs of Commutative Rings
Let R be a commutative ring with nonzero identity and I an ideal of R. The focus of this research is on a generalization of the zero-divisor graph called the ideal-based zero-divisor graph for commutative rings with nonzero identity. We consider such a graph to be nontrivial when it is nonempty and distinct from the zero-divisor graph of R. We begin by classifying all rings which have nontrivial ideal-based zero-divisor graph complete on fewer than 5 vertices. We also classify when such graphs are complete on a prime number of vertices. In addition we classify all rings which admit nontrivial planar ideal-based zero-divisor graph. The ideas of complemented and uniquely complemented are considered for such graphs, and we classify when they are uniquely complemented. The relationship between graph isomorphisms of the ideal-based zero divisor graph with respect to I and graph isomorphisms of the zero-divisor graph of R/I [R mod I] is also considered. In the later chapters, we consider properties of ideal-based zero-divisor graphs when the corresponding factor rings are Boolean or reduced. We conclude by giving all nontrivial ideal based zero-divisor graphs on less than 8 vertices, a few miscellaneous results, and some questions for future research
The Auslander-Bridger formula and the Gorenstein property for coherent rings
The concept of Gorenstein dimension, defined by Auslander and Bridger for
finitely generated modules over a Noetherian ring, is studied in the context of
finitely presented modules over a coherent ring. A generalization of the
Auslander-Bridger formula is established and is used as a cornerstone in the
development of a theory of coherent Gorenstein rings.Comment: 28 page
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