Classification of non-local rings with projective 3-zero-divisor hypergraph

Let R be a commutative ring with identity and let Z(R, k) be the set of all k-zero-divisors in R and $$k>2$$ an integer. The k-zero-divisor hypergraph of R, denoted by $$\mathcal {H}_k(R)$$ , is a hypergraph with vertex set Z(R, k), and for distinct elements $$x_1,x_2,\ldots ,x_k$$ in Z(R, k), the set $$\{x_1,x_2, \ldots , x_k\}$$ is an edge of $$\mathcal {H}_k(R)$$ if and only if $$\prod \limits _{i=1}^kx_i =0$$ and the product of any $$(k-1)$$ elements of $$\{x_1,x_2,\ldots ,x_k\}$$ is nonzero. In this paper, we characterize all finite commutative non-local rings R with identity whose $$\mathcal {H}_3(R)$$ has crosscap one.