On the line graph structure of the cozero-divisor graph of a commutative ring

Assume that R is a commutative ring with non-zero identity and \(W^*(R)\) $W^{\ast }(R)$ is the set of all non-zero non-unit elements of R. Also, for \(x\in R\) $x\in R$ , the ideal which is generated by x, is denoted by Rx. The cozero-divisor graph of R, which is denoted by \(\Gamma '(R)\) ${\mathrm{\Gamma }}^{\prime }(R)$ , is a graph with \(W^*(R)\) $W^{\ast }(R)$ as the vertex-set, and two distinct vertices x and y are adjacent in \(W^*(R)\) $W^{\ast }(R)$ if and only if \(x\notin Ry\) $x\notin Ry$ and \(y\notin Rx\) $y\notin Rx$ . In this paper, we completely determine all finite commutative rings R such that \(\Gamma '(R)\) ${\mathrm{\Gamma }}^{\prime }(R)$ is a line graph. We also characterize all finite commutative rings R such that \(\Gamma '(R)\) ${\mathrm{\Gamma }}^{\prime }(R)$ is isomorphic to its line graph.