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گراف مقسوم علیه های صفر و گراف تام حلقه ی جابه جایی

zero-divisor graph and total graph of commutative ring

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  1. Tez No: 775434
  2. Yazar: ROGHAYEH HAFEZIEH
  3. Danışmanlar: PROF. DR. MOHAMMADTAGHI DIBAEI
  4. Tez Türü: Yüksek Lisans
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 2008
  8. Dil: Farsça
  9. Üniversite: Kharazmi University
  10. Enstitü: Yurtdışı Enstitü
  11. Ana Bilim Dalı: Matematik Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 113

Özet

Let $R$ be a commutative ring with $Nil(R)$ its ideal of nilpotent elements, $Z(R)$ its set of zero-divisors, $Reg(R)$ its set of regular elements, and $Z(R)^*=Z(R)\backslash\{\;$0$\;\}$ its set of nonzero zero-divisors. Zero-divisor graph of $R$ is (an undirected) graph with $Z(R)^*$ as vertices, such that for each $x,y\in Z(R)^*$ ($x\neq y$), the vertices $x$ and $y$ are adjacent if and only if $xy=\;$0. We denote this graph by $\Gamma(R)$. In this discussion we charactrize the conditions in which $diam(\Gamma(R))\leq\;$2 or $gr(\Gamma(R))\geq\;$4, and then use the results for polynomial rings, power series rings, and idealizations. Also the total graph of $R$ is a graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. We show this graph by $T(\Gamma(R))$. We also study the subgraphs $Nil(\Gamma(R))$, $Z(\Gamma(R))$, and $Reg(\Gamma(R))$ of $T(\Gamma(R))$, with vertices $Nil(R)$, $Z(R)$, and $Reg(R)$, respectively.

Özet (Çeviri)

Let $R$ be a commutative ring with $Nil(R)$ its ideal of nilpotent elements, $Z(R)$ its set of zero-divisors, $Reg(R)$ its set of regular elements, and $Z(R)^*=Z(R)\backslash\{\;$0$\;\}$ its set of nonzero zero-divisors. Zero-divisor graph of $R$ is (an undirected) graph with $Z(R)^*$ as vertices, such that for each $x,y\in Z(R)^*$ ($x\neq y$), the vertices $x$ and $y$ are adjacent if and only if $xy=\;$0. We denote this graph by $\Gamma(R)$. In this discussion we charactrize the conditions in which $diam(\Gamma(R))\leq\;$2 or $gr(\Gamma(R))\geq\;$4, and then use the results for polynomial rings, power series rings, and idealizations. Also the total graph of $R$ is a graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. We show this graph by $T(\Gamma(R))$. We also study the subgraphs $Nil(\Gamma(R))$, $Z(\Gamma(R))$, and $Reg(\Gamma(R))$ of $T(\Gamma(R))$, with vertices $Nil(R)$, $Z(R)$, and $Reg(R)$, respectively.

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