Toroidal zero-divisor graphs of decomposable commutative rings without identity

Let R be a commutative ring without identity. The zero-divisor graph of R,  denoted by Γ(R),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma (R),$$\end{document} is a graph with vertex set Z(R)\{0},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(R){{\setminus }} \{0\},$$\end{document} which is the set of all non-zero zero-divisor elements of R and two vertices x and y are adjacent if and only if xy=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$xy=0.$$\end{document} In this paper, we characterize (up to isomorphism) all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal.