Remark 2

Suppose that M is a direct sum of two submodules N and I that are stable under $u$ and that $u$ induces a nilpotent endomorphism $u_N$ of N and an automorphism of I. We then have $N_{\infty}= N$ and $I_{\infty}= I$, and the sequences $(N_p)$ and $(I_p)$ are stationary. Moreover, the following integers are equal:

$\alpha )$ the least integer $p\geqslant 0$ such that $N_p= N_{\infty}$,

$\beta )$ the least integer $q\geqslant 0$ such that $I_q= I_{\infty}$,

$\gamma )$ the least integer $r\geqslant 0$ such that $(u_N)^r= 0$.