Proposition 4

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

We write $v = T(u): T(M) \to T(N)$; we know (§ 5, no. 2, Proposition 3) that $v$ is surjective and hence it follows from the definitions that $v(\mathfrak{J}'_M) = \mathfrak{J}'_N$; if $\mathfrak{R}$ is the kernel of $v$, then $v^{-1}(\mathfrak{J}'_N) = \mathfrak{R} + \mathfrak{J}'_M$. As $S(u): T(M)/\mathfrak{J}'_M \to T(N)/\mathfrak{J}'_N$ is derived from $v$ by passing to the quotients, it is a surjective homomorphism whose kernel is $\mathfrak{R}' = (\mathfrak{R} + \mathfrak{J}'_M)/\mathfrak{J}'_M$. As $\mathfrak{R}$ is generated by the kernel $P$ of $u$ (§ 5, no. 2), so is $\mathfrak{R}'$.

If $u: M \to N$ is an injective linear mapping, it is not always true that $S(u)$ is an injective mapping (Exercise 1). However it is so when $u$ is an injection such that $u(M)$ is a direct factor in $N$ and then the image of $S(u)$ (isomorphic to $S(M)$) is a direct factor of $S(N)$; the proof is the same as that for the analogous assertions for $T(u)$ (§ 5, no. 2) replacing $T$ by $S$.