The exact zero-divisor graph of a reduced ring

S. Visweswaran () and Premkumar T. Lalchandani ()
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S. Visweswaran: Saurashtra University
Premkumar T. Lalchandani: Faculty, Department of Mathematics, Dr. Subhash Science College

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 4, 1123-1144

Abstract: Abstract The rings considered in this article are commutative with identity which are not integral domains. Let R be a ring. Recall that an element x of R is an exact zero-divisor if there exists a non-zero element y of R such that $$Ann(x) = Ry$$ A n n ( x ) = R y and $$Ann(y) = Rx$$ A n n ( y ) = R x . As in Lalchandani (International J. Science Engineering and Management (IJSEM) 1(6): 14-17, 2016), for a ring R, we denote the set of all exact zero-divisors of R by EZ(R) and $$EZ(R)\backslash \{0\}$$ E Z ( R ) \ { 0 } by $$EZ(R)^{*}$$ E Z ( R ) ∗ . Let R be a ring. In the above mentioned article, Lalchandani introduced and studied the properties of a graph denoted by $$E\Gamma (R)$$ E Γ ( R ) , which is an undirected graph whose vertex set is $$EZ(R)^{*}$$ E Z ( R ) ∗ and distinct vertices x and y are adjacent in $$E\Gamma (R)$$ E Γ ( R ) if and only if $$Ann(x) = Ry$$ A n n ( x ) = R y and $$Ann(y) = Rx$$ A n n ( y ) = R x . Let R be a reduced ring such that $$EZ(R)^{*}\ne \emptyset $$ E Z ( R ) ∗ ≠ ∅ . The aim of this article is to study the interplay between the graph-theoretic properties of $$E\Gamma (R)$$ E Γ ( R ) and the ring-theoretic properties of R.

Keywords: Reduced ring; von Neumann regular ring; Exact zero-divisor; Girth; Clique number; Chromatic number; Primary; :13A15; and; Secondary:; 05C25 (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1007/s13226-021-00086-9

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