Is the quotient of a torsion-free module torsion-free?

Question

Let $R=\mathbf{Z}[G]$, where $G$ is a finite group (not necessarily commutative). Suppose further that $M$ is an $R$-mod which is torsion-free. Can I say anything about the torsion elements of $M/R^n$ where $n\geq 1$? Is this torsion-free? If so, then why?

Answer 1

Original poster said in comments that “wanted to quotient by any free module”. Then the problem is already solved by @Nishant: the quotient is not necessarily torsion-free.

This follows from:

• The submodule m Z of Z is free 1-dimensional (if m ≠ 0), and Z / m Z has torsion iff |m| > 1;
• • In fact, any non-trivial torsion-free Z-module contains a 1-dimenional free submodule (isomorphic to Z), and likewise quotients with torsion;
• If G is trivial, then the group ring Z[G] is the same as Z.