Total graphs associated to a commutative ring. (English) Zbl 1346.13005
Summary: Let \(R\) be a commutative ring with nonzero unity. Let \(Z(R)\) be the set of all zero-divisors of \(R\). The total graph of \(R\), denoted by \(T(\Gamma(R))\), is the simple graph with vertex set \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if their sum \(x+y \in Z(R)\). Several authors presented various generalizations for \(T(\Gamma(R))\). This article surveys research conducted on \(T(\Gamma(R))\) and its generalizations. A historical review of literature is given. Further properties of \(T(\Gamma(R))\) are also studied. Many open problems are presented for further research.
MSC:
13A15 | Ideals and multiplicative ideal theory in commutative rings |
13B99 | Commutative ring extensions and related topics |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |