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Using Composition--Diamond Lemma we construct presentations of groups $G =\langle x_1,\ldots,x_n \, | \, r_1,\ldots, r_m \rangle$ with the followingproperty; for a fixed $1 \le i \le n$, and for all $1 \le j \le m$, Foxderivatives $\partial r_j / \partial x_i$ have common divisor. It follows thatif $\pi_2(K) \ne 0$, where $K$ is the standard $2$-complex associated with $G$then the group ring $\mathbb{Z}[G]$ has nontrivial zero divisors.

On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings