• Title/Summary/Keyword: zero divisor module

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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.

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.

A GENERALIZATION OF THE ZERO-DIVISOR GRAPH FOR MODULES

  • Safaeeyan, Saeed;Baziar, Mohammad;Momtahan, Ehsan
    • Journal of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.87-98
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    • 2014
  • Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

ON THE 2-ABSORBING SUBMODULES AND ZERO-DIVISOR GRAPH OF EQUIVALENCE CLASSES OF ZERO DIVISORS

  • Shiroyeh Payrovi;Yasaman Sadatrasul
    • Communications of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.39-46
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    • 2023
  • Let R be a commutative ring, M be a Noetherian R-module, and N a 2-absorbing submodule of M such that r(N :R M) = 𝖕 is a prime ideal of R. The main result of the paper states that if N = Q1 ∩ ⋯ ∩ Qn with r(Qi :R M) = 𝖕i, for i = 1, . . . , n, is a minimal primary decomposition of N, then the following statements are true. (i) 𝖕 = 𝖕k for some 1 ≤ k ≤ n. (ii) For each j = 1, . . . , n there exists mj ∈ M such that 𝖕j = (N :R mj). (iii) For each i, j = 1, . . . , n either 𝖕i ⊆ 𝖕j or 𝖕j ⊆ 𝖕i. Let ΓE(M) denote the zero-divisor graph of equivalence classes of zero divisors of M. It is shown that {Q1∩ ⋯ ∩Qn-1, Q1∩ ⋯ ∩Qn-2, . . . , Q1} is an independent subset of V (ΓE(M)), whenever the zero submodule of M is a 2-absorbing submodule and Q1 ∩ ⋯ ∩ Qn = 0 is its minimal primary decomposition. Furthermore, it is proved that ΓE(M)[(0 :R M)], the induced subgraph of ΓE(M) by (0 :R M), is complete.

STRUCTURE OF THE FLAT COVERS OF ARTINIAN MODULES

  • Payrovi, S.H.
    • Journal of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.611-620
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    • 2002
  • The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form $\prod_{p\epsilonAtt_R(A)}T-p$, where $Tp$ is the completion of a free R$_{p}$-module. Also, we construct a minimal flat resolution for R/xR-module 0: $_AX$ from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = $\chiA$. This result leads to a description of the structure of a minimal flat resolution for ${H^n}_{\underline{m}}(R)$, nth local cohomology module of R with respect to the ideal $\underline{m}$, over a local Cohen-Macaulay ring (R, $\underline{m}$) of dimension n.

ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS

  • Pirzada, Shariefuddin;Raja, Rameez
    • Journal of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1167-1182
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    • 2016
  • Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.

A TORSION GRAPH DETERMINED BY EQUIVALENCE CLASSES OF TORSION ELEMENTS AND ASSOCIATED PRIME IDEALS

  • Reza Nekooei;Zahra Pourshafiey
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.797-811
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    • 2024
  • In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by AE(M). The vertex set of AE(M) is the set of equivalence classes {[x] | x ∈ T(M)*}, where two torsion elements x, y ∈ T(M)* are equivalent if ann(x) = ann(y). Also, two distinct classes [x] and [y] are adjacent in AE(M), provided that ann(x)ann(y)M = 0. We shall prove that for every torsion finitely generated module M over a Dedekind domain R, a vertex of AE(M) has degree two if and only if it is an associated prime of M.