On the Planar Property of an Ideal-Based Weakly Zero-Divisor Graph

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Let R be a commutative ring with a nonzero identity and Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R, denoted by WГ(R), is the graph with the vertex set 〖Z(R)〗^*=Z(R)\\{0}, where two distinct vertices a and b form an edge if ar=bs=rs=0 for r,s∈R\\{0}. For an ideal I of R, the ideal-based zero-divisor graph of R, denoted by Г_I (R), has vertices {a∈R\I:ab∈I for some b∈R\I} and edges {(a,b):ab∈I,a,b∈R\I,a≠b}. In this article, an ideal-based weakly zero-divisor graph of R, denoted by 〖WГ〗_I (R), is introduced which contains Г_I (R) as a subgraph and is identical to the graph WГ(R) when I={0}. The relationship between the graphs WГ_I (R) and WГ(R/I) is investigated and the planar property of 〖WГ〗_I (R) is studied. The results show that WГ(R/I) is isomorphic to a subgraph of 〖WГ〗_I (R). For 〖WГ〗_I (R) to be planar, some restraints are provided on the size of the ideal I and girth of 〖WГ〗_I (R). In conclusion, the results suggest that WГ_I (R) and WГ(R/I) are strongly related and establish necessary and sufficient conditions for WГ_I (R) to be planar. In addition, rings R with planar WГ_I (R) are classified.