Article in volume
Authors:
Hiren D. Patel
Faculty, Government Polytechnic
Bhuj-370001, Gujarat, India
email: hdp12376@gmail.com
Title:
Some remarks on the complement of the Armendariz graph of a commutative ring
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 43(1) (2023) 5-24
Received: 2021-02-04 , Revised: 2021-05-13 , Accepted: 2021-05-19 , Available online: 2022-11-29 , https://doi.org/10.7151/dmgaa.1403
Abstract:
Let $R$ be a commutative ring with identity which is not an integral domain.
Let $Z(R)$ denote the set of all zero-divisors of $R$. Recall from [1]
that the Armendariz graph of $R$ denoted by $A(R)$ is an undirected graph whose
vertex set is $Z(R[X])\backslash \{0\}$ and distinct vertices $f(X) =
\sum_{i = 0}^{n}a_{i}X^{i}$ and $g(X) = \sum_{j = 0}^{m}b_{j}X^{j}$ are adjacent
in $A(R)$ if and only if $a_{i}b_{j} = 0$ for all $i\in \{0, \ldots, n\}$ and
$j\in \{0, \ldots, m\}$. The aim of this article is to study the interplay
between the graph-theoretic properties of the complement of $A(R)$, that is,
$(A(R))^{c}$ and the ring-theoretic properties of $R$.
Keywords:
B-prime of $(0)$, complement of the zero-divisor graph, diameter, domination number, maximal N-prime of $(0)$, radius
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