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ABSTRACT
Let R be a commutative ring and Z(R)* be its set of all nonzero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by Badawi [Citation8]. In this paper, we classify the finite commutative rings whose AG(R) are projective. Also we determine all isomorphism classes of finite commutative rings with identity whose AG(R) has genus two.
Acknowledgments
The authors would like to thank the referee for careful reading of the manuscript and helpful comments. The work reported here is supported by the UGC Major Research Project (F. No. 42-8/2013(SR)) awarded to the first author by the University Grants Commission, Government of India. Also the work is supported by the INSPIRE programme (IF 140700) of Department of Science and Technology, Government of India for the second author.