Let $\Lambda_d := \mathbb{Z}_p[[T_1, \ldots, T_d]]$ denote the ring of formal power series in $d$ variables over the ring of $p$-adic integers. Suppose that $g \in \Lambda_d$ is an irreducible element, let $n$ be a positive natural number, and define $$E := \Lambda_d/(g^n).$$ Let $A \subseteq E$ be a $\Lambda_d$-submodule, i.e., $A = C/(g^n)$ for some ideal $C$ of $\Lambda_d$ containing $g^n$. We assume that $C$ contains two coprime elements.
Let now $f_1, \ldots, f_d$ denote monic polynomials in $\mathbb{Z}_p[T_1], \ldots, \mathbb{Z}_p[T_d]$, respectively.
I would like to prove the following statement:
$E/((f_1, \ldots, f_d) \cdot E)$ is finite if and only if $A/((f_1, \ldots, f_d) \cdot A)$ is finite, and in this case
$$ |A/((f_1, \ldots, f_d) \cdot A)| \; \ge \; |E/((f_1, \ldots, f_d) \cdot E)| .$$
If $d = 1$, then one can show that in fact
$$ |A/((f_1, \ldots, f_d) \cdot A)| \; = \; |E/((f_1, \ldots, f_d) \cdot E)| .$$ This is basically an application of the Snake Lemma.
For $d > 1$, however, there exist examples where the order of $|A/((f_1, \ldots, f_d) \cdot A)|$ is strictly larger. For example, let $d = 3$, $g = p$ and $f_i = T_i$, respectively. Let $A := (p,T_1,T_2)/(p) \subseteq E$.
Then $|A/((f_1, f_2,f_3) \cdot A)| = 2$, whereas $|E/((f_1, f_2, f_3) \cdot E)| = 1$.
I have not been able to prove the inequality even in the special case $f_i = T_i$, $1 \le i \le d$.
Thank you in advance!
