A cancellation theorem for metacyclic group rings

A ring \(\Lambda \) $\mathrm{\Lambda }$ has stably free cancellation when every stably free \(\Lambda \) $\mathrm{\Lambda }$ -module is free. Let \( G \; = \; C_p \rtimes C_q \) $G=C_p⋊C_q$ be a finite metacyclic group where p is an odd prime and q is a positive integral divisor of \(p-1\) $p-1$ . We show that the group ring \(\mathcal{R}[G]\) $\mathcal{R}[G]$ has stably free cancellation when \(\;\mathcal{R} \; = \; {\mathbb {Z}}[t_1, t_1^{-1}, \dots t_m, t_m^{-1}, x_1, \dots x_n] \;\) $\mathcal{R}=\mathbb{Z}[t_1,t_1^{-1},\cdots t_m,t_m^{-1},x_1,\cdots x_n]$ is a ring of mixed polynomials and Laurent polynomials over the integers. As a consequence, when \(C_\infty ^{(m)}\) $C_{\infty }^{(m)}$ is the free abelian group of rank m then the integral group ring \({\mathbb {Z}}[G(p,q) \times C_\infty ^{(m)}]\; \) $\mathbb{Z}[G(p,q)\times C_{\infty }^{(m)}]$ has stably free cancellation.