Remark

Note in particular that the hypotheses of Proposition 4 always hold for arbitrary submodules $N, P$ of $M$ when $A$ is a field (II, § 7, no. 3, Proposition 4). Moreover, if $N \subset P$ and $N \neq P$, then $T^n(N) \neq T^n(P)$ for all $n \geq 1$, since if $R$ is a complement of $P$ in $N$, then $T^n(P) \cap T^n(R) = \{0\}$ by (4) and $T^n(R) \neq \{0\}$.