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

Abstract

Using Composition--Diamond Lemma we construct presentations of groups $G=⟨x_1,\dots ,x_n|r_1,\dots ,r_m⟩$ 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.