A cancellation theorem for metacyclic group rings

Abstract

AbstractA ring $$\Lambda $$ Λ has stably free cancellation when every stably free $$\Lambda $$ Λ -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 $$\mathcal{R}[G]$$ R [ G ] has stably free cancellation when $$\;\mathcal{R} \; = \; {\mathbb {Z}}[t_1, t_1^{-1}, \dots t_m, t_m^{-1}, x_1, \dots x_n] \;$$ R = Z [ t 1 , t 1 - 1 , ⋯ t m , t m - 1 , x 1 , ⋯ x n ] is a ring of mixed polynomials and Laurent polynomials over the integers. As a consequence, when $$C_\infty ^{(m)}$$ C ∞ ( m ) is the free abelian group of rank m then the integral group ring $${\mathbb {Z}}[G(p,q) \times C_\infty ^{(m)}]\; $$ Z [ G ( p , q ) × C ∞ ( m ) ] has stably free cancellation.