Proposition 3

If $u : M \to N$ is a surjective A-linear mapping, the homomorphism $\mathbf{T}(u) : \mathbf{T}(M) \to \mathbf{T}(N)$ is surjective and its kernel is the two-sided ideal of $\mathbf{T}(M)$ generated by the kernel $P \subset M \subset \mathbf{T}(M)$ of $u$.

$\mathbf{T}^0(u) : \mathbf{T}^0(M) \to \mathbf{T}^0(N)$ is bijective and for every integer $n > 0$,

is surjective, as is seen by induction on $n$ using II, § 3, no. 6, Proposition 6; the latter proposition also shows, by induction on $n$, that the kernel $\mathfrak{I}_n$ of $T^n(u)$ is the submodule of $T^n(M)$ generated by the products

where at least one of the $x_i$ belongs to $P$. This shows that the kernel $\mathfrak{I} = \bigoplus_{n \geq 1} \mathfrak{I}_n$ of $T(u)$ is the two-sided ideal generated by $P$ in $T(M)$.

If $u : M \to N$ is an injective linear mapping, it is not always true that $T(u)$ is an injective mapping (Exercise 1). However, this is true when $u$ is an injection such that $u(M)$ is a direct factor of $N$, for then there exists a linear mapping $v : N \to M$ such that $v \circ u$ is the identity mapping on $M$ and therefore

is the identity mapping on $T(M)$, hence $T(u)$ is injective and its image (isomorphic to $T(M)$) is a direct factor of $T(N)$ (II, § 1, no. 9, Proposition 15). More precisely: