The total graph of a commutative ring

• Main Authors: Ting-Yi zhao, 趙庭億
• Other Authors: Hsin-Ju Wang
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/93108078925853095794

碩士 === 國立中正大學 === 應用數學研究所 === 101 === Let $R$ be a commutative ring. We use $Z(R)$, $\Reg(R)$ and $\Nil(R)$ to indicate the sets of zero-divisors, regular elements and nilpotent elements in $R$, respectively. In this thesis, we shall introduce and investigate the \emph{total graph} $T(\Gamma(R))$ of $R$. It is the (undirected) 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 $x+y \in Z(R)$. It is an important problem for us to discuss the planarity of $T(\Gamma(R))$. We also study the three subgraphs $Z(\Gamma(R))$, $\Reg(\Gamma(R))$, and $\Nil(\Gamma(R))$ of $T(\Gamma(R))$, with vertices $Z(R)$, $\Reg(R)$, and $\Nil(R)$, respectively.