For a commutative ring R with non-zero identity, we denote the set of non-zero element x in R with xR neq R by W^*(R). So W^*(R) is the set of non-units of R. In this paper, we introduced the cozero-divisor graph Gamma\\\'(R), which is a dual of zero-divisor graph Gamma(R) in some sense, as the (undirected) graph with vertices W^*(R) and for two distinct elements x and y in W^*(R), the vertices x and y are adjacent if and only if x not in R and y not in xR. We investigate the interplay between the ring-theoretic properties of R and graph-theoretic properties of Gamma\\\\\\\'(R). Also we study the relations between two graphs Gamma(R) and Gamma\\\'(R).

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