• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1409-1418
• /
• 2021

In this paper, we study domination in the zero-divisor graph of a *-ring. We first determine the domination number, the total domination number, and the connected domination number for the zero-divisor graph of the product of two *-rings with componentwise involution. Then, we study domination in the zero-divisor graph of a Rickart *-ring and relate it with the clique of the zero-divisor graph of a Rickart *-ring.

• Honam Mathematical Journal
• /
• v.34 no.4
• /
• pp.571-584
• /
• 2012

In this study, we investigate the concept of zero-divisor graphs of multiplication modules over commutative rings as a natural generalization of zero-divisor graphs of commutative rings. In particular, we study the zero-divisor graphs of the module $\mathbb{Z}_n$ over the ring $\mathbb{Z}$ of integers, where $n$ is a positive integer greater than 1.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.6
• /
• pp.1851-1861
• /
• 2014

We introduce the concept of w-zero-divisor (w-ZD) rings and study its related rings. In particular it is shown that an integral domain R is an SM domain if and only if R is a w-locally Noetherian w-ZD ring and that a commutative ring R is w-Noetherian if and only if the polynomial ring in one indeterminate R[X] is a w-ZD ring. Finally we characterize universally zero divisor rings in terms of w-ZD modules.

• Kyungpook Mathematical Journal
• /
• v.62 no.3
• /
• pp.595-613
• /
• 2022

Let R be a commutative ring with identity and let I be a proper ideal of R. In this paper, we study the ideal based extended zero-divisor graph 𝚪'_I (R) and prove that 𝚪'_I (R) is connected with diameter at most two and if 𝚪'_I (R) contains a cycle, then girth is at most four girth at most four. Furthermore, we study affinity the connection between the ideal based extended zero-divisor graph 𝚪'_I (R) and the ideal-based zero-divisor graph 𝚪_I (R) associated with the ideal I of R. Among the other things, for a radical ideal of a ring R, we show that the ideal-based extended zero-divisor graph 𝚪'_I (R) is identical to the ideal-based zero-divisor graph 𝚪_I (R) if and only if R has exactly two minimal prime-ideals which contain I.

• Journal of the Korean Mathematical Society
• /
• v.46 no.2
• /
• pp.313-325
• /
• 2009

We consider zero-divisor graphs with respect to primal, nonprimal, weakly prime and weakly primal ideals of a commutative ring R with non-zero identity. We investigate the interplay between the ringtheoretic properties of R and the graph-theoretic properties of ${\Gamma}_I(R)$ for some ideal I of R. Also we show that the zero-divisor graph with respect to primal ideals commutes by localization.

• Communications of the Korean Mathematical Society
• /
• v.32 no.1
• /
• pp.7-17
• /
• 2017

Let R be a commutative ring with zero-divisors Z(R). The extended zero-divisor graph of R, denoted by $\bar{\Gamma}(R)$, is the (simple) graph with vertices $Z(R)^*=Z(R){\backslash}\{0\}$, the set of nonzero zero-divisors of R, where two distinct nonzero zero-divisors x and y are adjacent whenever there exist two non-negative integers n and m such that $x^ny^m=0$ with $x^n{\neq}0$ and $y^m{\neq}0$. In this paper, we consider the extended zero-divisor graphs of idealizations $R{\ltimes}M$ (where M is an R-module). At first, we distinguish when $\bar{\Gamma}(R{\ltimes}M)$ and the classical zero-divisor graph ${\Gamma}(R{\ltimes}M)$ coincide. Various examples in this context are given. Among other things, the diameter and the girth of $\bar{\Gamma}(R{\ltimes}M)$ are also studied.

• Korean Journal of Mathematics
• /
• v.29 no.1
• /
• pp.13-24
• /
• 2021

For a finite commutative ring R with identity 1 ≠ 0, the zero divisor graph ��(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs ��(ℤn) for various values of n. Also, we find signless Laplacian spectrum of ��(ℤn) for n = pz, z ≥ 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph ��(ℤn) are signless Laplacian integral.

• Communications of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.391-399
• /
• 2019

Let R be a commutative ring with identity and let M be an R-module. The main purpose of this paper is to calculate the chromatic number of the zero-divisor graphs over modules.

• Journal of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1403-1418
• /
• 2019

Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that $f(x)=c_f f_1(x)$ with $c_f{\in}R$ and $f_1(x)$ is not zero-divisor, then $c_f$ is called an annihilating content for f(x). In this case $Ann(f)=Ann(c_f )$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs ${\Gamma}(R)$ and ${\Gamma}(R[x])$ are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

• Korean Journal of Mathematics
• /
• v.27 no.3
• /
• pp.563-580
• /
• 2019

In this paper, we study the strong metric dimension of zero-divisor graph ${\Gamma}(R)$ associated to a ring R. This is done by transforming the problem into a more well-known problem of finding the vertex cover number ${\alpha}(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring ${\mathbb{Z}}_n$ of integers modulo n and the ring of Gaussian integers ${\mathbb{Z}}_n$[i] modulo n. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.