The total graph of a commutative ring
Let \(R\) be a commutative ring with \(Nil(R)\) its ideal of nilpotent elements, \(Z(R)\) its set of zero-divisors, and \(Reg(R)\) its set regular elements. Let \(T(\Gamma(R))\) be the total graph of \(R\), \(Reg(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Reg(R)\), \(Z(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Z(R)\) and \(Nil(\Gamma(R))\) be the subgraph of \(T(\Gamma(R))\) with vertices \(Nil(R)\). The main goal of the first section to provide general structure theorem for \(Reg(\Gamma(R))\), \(Z(\Gamma (R))\) and \(Nil(\Gamma(R))\), in case where \(Z(R)\) is an ideal of \(R\), \(Z(\Gamma(R))\) is a complete subgraph of \(T(\Gamma(R))\) and is disjoint from \(Reg(\Gamma(R))\). In the first section, the authors complete the diameter and the girth of \(Reg(\Gamma(R))\), and show that \(Z(\Gamma(R))\) is always connected and \(T(\Gamma(R))\) is never connected if \(Z(R)\) is an ideal. Finally, the basic of properties of \(Nil(\Gamma(R))\) are given below: Let \(R\) be a commutative ring: (1) \(Nil(\Gamma(R))\) is a complete (induced) subgraph of \(Z(\Gamma(R))\); (2) Each vertex of \(Nil(\Gamma(R))\) is adjacent to each distinct vertex; (3) \(Nil(\Gamma(R))\) is disjoint from \(Reg(\Gamma(R))\); (4) If \(\{0\}\neq Nil(R)\subseteq Z(R)\), then \(gr(Z(\Gamma(R)))=3\). Section 2 traits the second case of \(Z(R)\), i.e., when \(Z(R)\) is not an ideal of \(R\). It shows that \(Z(\Gamma(R))\) is always connected but never complete, \(Z(\Gamma(R))\) and \(Reg(\Gamma(R))\) are never disjoint subgraphs of \(T(\Gamma(R))\) and \(|Z(R)|>3\). The authors conclude this section by the investigation of the girth of \(Z(\Gamma(R))\), \(Reg(\Gamma(R))\) and \(T(\Gamma(R))\) when \(Z(R)\) is not an ideal of \(R\).
scientific article; zbMATH DE number 5376932
scientific article