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

Abstract

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