Let $M$ be an $A$-module, and suppose that the length $\ell (M)<\mathrm{\infty }$ (that is, a composition series exists). Then the following statements hold,

1. If $N\le M$, then 1;;$\ell (N)<\ell (M)$
2. If $N<M$, then 1;;$\ell (N)<\ell (M)$
3. Any strictly increasing chain of submodules has length $n\le$ 1;;$\ell (M)$.
4. Every composition series has length $\ell (M)$.
5. Any strictly increasing chain of submodules 1;;extends to a composition series