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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.)