Abstract
Let R be a commutative ring with Z(R) the set of zero-divisors and U(R) the set of unit elements of R. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the domination number of T(Γ(R)). It is shown that if R = Z(R) ∪ U(R), then the domination number of T(∪(R)) is finite provided R has a maximal ideal of finite index. Moreover, if
The research of H. R. Maimani was in part supported by a grant from IPM (No. 93050112).
Acknowledgement
The authors are grateful to the anonymous referee for making many constructive suggestions. Part of this work was done while S. Yassemi was visited the Max Planck Institute for Mathematics (MPIM), Bonn, Germany. He would like to thank MPIM for sponsoring his visit to Bonn in 2014.
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