Let $H$ be a subgroup of $G$. Then a homomorphism $r:G\to H$ is said to be a retraction if the inclusion homomorphism $i:H\hookrightarrow G$ is a right inverse of $r$, i.e. $r(x)=x$ for all elements $x\in H$. Then $H$ is called a retract of $G$ and denoted by $H<_r G$ (we can define retracts of $R$-modules similarly, which I call $R$-retracts).
A group $G$ is said to satisfy the descending chain conditions on retracts if for every chain of $G_1 >_r G_2 >_r \cdots$ of retracts of $G$ there is an integer $n$ such that $G_i = G_n$ for all $i\geq n$.
My question : Assume that $G$ satisfies the descending chain conditions on retracts and $M$ is a finitely generated $\mathbb{Z}G$-module. Does $M$ satisfy the descending chain conditions on $\mathbb{Z}G$-retracts?