Resumen de On the planarity, genus, and crosscap of the weakly zero-divisor graph of commutative rings

Nadeem ur Rehman, Mohd Nazim, Shabir Ahmad Mir

• Let R be a commutative ring and Z(R) its zero-divisors set. The weakly zero-divisor graph of R, denoted by WΓ(R) is an undirected graph with the nonzero zero-divisors Z(R)∗as vertex set and two distinct vertices x and y are adjacent if and only if there exist a∈Ann(x) and b∈Ann(y) such that ab=0. In this paper, we characterize finite rings R for which the weakly zero-divisor graph WΓ(R) belongs to some well-known families of graphs. Further, we classify the finite rings R for which WΓ(R) is planar, toroidal or double toroidal. Finally, we classify the finite rings R for which the graph WΓ(R) has crosscap at most two.