Proposition 2

Let $C$ and $C'$ be two $p$-rings and $u$ a homomorphism of $C$ into $C'$. Let $v$ be the homomorphism of $\kappa_C = C/pC$ into $\kappa_{C'} = C'/pC'$ deduced from $u$ by passing to quotients.

a) One has $l(C) \geqslant l(C')$, and $u$ is injective if and only if one has $l(C) = l(C')$.

b) In order that $u$ be surjective, it is necessary and sufficient that $v$ be an isomorphism.

c) In order that $u$ be an isomorphism, it is necessary and sufficient that $v$ be an isomorphism and that one have $l(C) = l(C')$.

Let $n \geqslant 1$ be an integer. We have $u(p^{n-1}1_C) = p^{n-1}1_{C'}$, hence the relation $p^{n-1}1_{C'} \neq 0$ implies $p^{n-1}1_C \neq 0$ and is equivalent to it if $u$ is injective. Therefore $l(C') \leqslant l(C)$, with equality if $u$ is injective. If $u$ is not injective, there exists an integer $i < l(C)$ such that the kernel of $u$ is the ideal $p^iC$ of $C$; then $p^i1_{C'} = 0$, whence $l(C') \leqslant i$. This proves a).

Since $\kappa_C$ and $\kappa_{C'}$ are fields, the homomorphism $v$ is injective. If $u$ is surjective, the same is true of $v$, which is therefore an isomorphism. Conversely, suppose $v$ surjective. Then for every integer $n \geqslant 0$, the mapping $v_n : p^nC/p^{n+1}C \to p^nC'/p^{n+1}C'$ induced by $u$ is surjective. Since $C$ is complete for the $pC$-adic filtration and $C'$ separated for the $pC'$-adic filtration, $u$ is surjective by Cor. 2 of Th. 1 of III, § 2, No. 8. This proves b).

Finally, c) follows from a) and b).