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The total graph of a commutative ring. (English) Zbl 1158.13001

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 \(R\).

MSC:

13A99 General commutative ring theory
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13B10 Morphisms of commutative rings
13C99 Theory of modules and ideals in commutative rings
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