• Title, Summary, Keyword: zero divisor

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ZERO-DIVISOR GRAPHS OF MULTIPLICATION MODULES

  • Lee, Sang Cheol;Varmazyar, Rezvan
    • Honam Mathematical Journal
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    • v.34 no.4
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    • pp.571-584
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    • 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.

A NOTE ON ZERO DIVISORS IN w-NOETHERIAN-LIKE RINGS

  • Kim, Hwankoo;Kwon, Tae In;Rhee, Min Surp
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1851-1861
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    • 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.

ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS

  • Atani, Shahabaddin Ebrahimi;Darani, Ahamd Yousefian
    • Journal of the Korean Mathematical Society
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    • v.46 no.2
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    • pp.313-325
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    • 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.

EXTENDED ZERO-DIVISOR GRAPHS OF IDEALIZATIONS

  • Bennis, Driss;Mikram, Jilali;Taraza, Fouad
    • Communications of the Korean Mathematical Society
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    • v.32 no.1
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    • pp.7-17
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    • 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.

ANNIHILATING CONTENT IN POLYNOMIAL AND POWER SERIES RINGS

  • Abuosba, Emad;Ghanem, Manal
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1403-1418
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    • 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.

ON STRONG METRIC DIMENSION OF ZERO-DIVISOR GRAPHS OF RINGS

  • Bhat, M. Imran;Pirzada, Shariefuddin
    • Korean Journal of Mathematics
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    • v.27 no.3
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    • pp.563-580
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    • 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.

ZERO DIVISOR GRAPHS OF SKEW GENERALIZED POWER SERIES RINGS

  • MOUSSAVI, AHMAD;PAYKAN, KAMAL
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.363-377
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    • 2015
  • Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.