scholarly journals The Graph of Equivalence Classes of Zero Divisors

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vinayak Joshi ◽  
B. N. Waphare ◽  
H. Y. Pourali
Keyword(s):  
Zero Divisor ◽  
Equivalence Classes ◽  
Zero Divisors ◽  
Boolean Lattices

We introduce a graph GE(L) of equivalence classes of zero divisors of a meet semilattice L with 0. The set of vertices of GE(L) are the equivalence classes of nonzero zero divisors of L and two vertices [x] and [y] are adjacent if and only if [x]∧[y]=[0]. It is proved that GE(L) is connected and either it contains a cycle of length 3 or GE(L)≅K2. It is known that two Boolean lattices L1 and L2 have isomorphic zero divisor graphs if and only if L1≅L2. This result is extended to the class of SSC meet semilattices. Finally, we show that Beck's Conjecture is true for GE(L) .

scholarly journals On Reduced Zero-Divisor Graphs of Posets

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Ashish Kumar Das ◽  
Deiborlang Nongsiang
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Equivalence Classes ◽  
Prime Ideals ◽  
Zero Divisors ◽  
Ascending Chain Condition

We study some properties of a graph which is constructed from the equivalence classes of nonzero zero-divisors determined by the annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.

scholarly journals On planarity of compressed zero-divisor graphs associated to commutative rings

2020 ◽  
Vol 29 (2) ◽  
pp. 131-136
Author(s):  
M. IMRAN BHAT ◽  
S. PIRZADA ◽  
AHMAD M. ALGHAMDI
Keyword(s):  
Equivalence Class ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Equivalence Classes ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The equivalence class [r] of an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re − {[0], [1]}, where Re = {[r] : r ∈ R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p 2 , p 3 and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn) and Γc(Zn[x]/(xm)) are planar.

scholarly journals Five-point zero-divisor graphs determined by equivalence classes

2011 ◽  
Vol 4 (1) ◽  
pp. 53-64
Author(s):  
Florida Levidiotis ◽  
Sandra Spiroff
Keyword(s):  
Zero Divisor ◽  
Equivalence Classes

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

Associate class graph of zero-divisors of a commutative ring

2019 ◽  
Vol 19 (08) ◽  
pp. 2050155
Author(s):  
Gaohua Tang ◽  
Guangke Lin ◽  
Yansheng Wu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
New Class ◽  
Associate Class ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Class Graph

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

scholarly journals On rings all of whose factor rings are integral domains

1993 ◽  
Vol 55 (3) ◽  
pp. 325-333
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Positive Integer ◽  
Zero Divisor ◽  
Arbitrary Positive Integer ◽  
Integral Domains ◽  
Factor Ring ◽  
Zero Divisors ◽  
Proper Quotient

AbstractA ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

scholarly journals When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Zero Divisors

Let be a commutative ring with identity which has at least two nonzero zero-divisors. Suppose that the complement of the zero-divisor graph of has at least one edge. Under the above assumptions on , it is shown in this paper that the complement of the zero-divisor graph of is complemented if and only if is isomorphic to as rings. Moreover, if is not isomorphic to as rings, then, it is shown that in the complement of the zero-divisor graph of , either no vertex admits a complement or there are exactly two vertices which admit a complement.

Characterizations of Three Classes of Zero-Divisor Graphs

2012 ◽  
Vol 55 (1) ◽  
pp. 127-137 ◽  
Author(s):  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Central Vertex ◽  
Integral Domains ◽  
Boolean Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

FINITE RINGS WITH EULERIAN ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250055 ◽  
Author(s):  
A. S. KUZMINA
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Noncommutative Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

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