We prove that the zero divisor problem is equivalent to some problem about subgroups of a free group.

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W ∗(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W ∗(R) are adjacent if and only if a / ∈ bR and b / ∈ aR. In this paper, we characterize all commutative rings whose cozero-divisor graphs are forest, star, doubl...