Groups $G$ for which every finitely generated $\mathbb{Z}G$-module is Hopfian

Question

Let $\mathbb{Z}G$ be the group ring over a group $G$. It is a well-known fact that every finitely generated module over a commutative ring is Hopfian. Hence if $G$ is abelian, then every finitely generated $\mathbb{Z}G$-module is Hopfian. On the other hand, P. Hall proved that if $G$ is a polycyclic-by-finite group and $R$ is right noetherian ring with identity, then the group ring $RG$ is right noetherian. Hence if $G$ is polycyclic-by-finite, then every finitely generated $ZG$-module is noetherian (hence Hopfian).

The only groups which are known to have a Noetherian group ring are polycyclic-by-finite. But:

Are there other classes of groups $G$ for which every finitely generated $\mathbb{Z}G$-module is Hopfian?

Answer 1

I'll show that if $F$ is a free group of rank at least three, then $\mathbb{Z}F$ has a finitely generated non-Hopfian module. (So this is not an answer to the question, but it seems relevant and is too long for a comment.)

Suppose $R$ is any ring with a finitely generated non-Hopfian right module $M$.

Then the ring $M_2(R)$ of $2\times2$ matrices over $R$ is another example, because it's Morita equivalent to $R$ (or, more explicitly, the $M_2(R)$-module $(M\text{ }M)$ of row vectors with entries from $M$ is a finitely generated non-Hopfian $M_2(R)$-module).

Also $M_2(R)$ is generated as a ring by units: If $X=\{x_i\vert i\in I\}$ is a generating set for $R$ in the strong sense that there is not even a proper nonunital subring of $R$ containing $X$, then it is easy to check that $$Y=\left\{\pmatrix{x_i&1\\1&0}\middle\vert i\in I\right\}\cup\left\{\pmatrix{0&1\\1&0}\right\}$$ is a generating set for $M_2(R)$ consisting of units.

So if $F$ is the free group on the set $Y$, there is a natural surjective ring homomorphism $\mathbb{Z}F\to M_2(R)$, and so we can regard $(M\text{ }M)$ as a finitely generated non-Hopfian right $\mathbb{Z}F$-module.

If we take $R$ to be the ring $$R=\mathbb{Z}\left\langle x,y\middle\vert xy=1\right\rangle,$$ then the maps given by left multiplication by $y$ and $x$ exhibit the free right module $R$ as a proper direct summand of itself, and so $R$ is a non-Hopfian module for itself. Since $R$ is generated by two elements, $M_2(R)$ is generated by three units, and so is a quotient of $\mathbb{Z}F$ for $F$ free of rank three.