Let $(V,T)$ with $T \in End_{\mathbb{k}}V$ be an $n$-dimensional $\mathbb{k}[X]$-module. If the minimal polynomial $m_T$ has an irreducible factorization $f_1 \cdots f_r$, can we say that:
$$ V \simeq \bigoplus_{i = 1}^r\frac{\mathbb{k}[X]}{(f_i)} \simeq \frac{(\mathbb{k}[X])^r}{(m_T)} $$
and if so, can we construct an explicit isomorphism?
I know that since $\operatorname{Ann}_{\mathbb{k}[X]}V = (m_T)$ we have that
$$ V \simeq \bigoplus_{i = 1}^rV[f_i] \simeq \bigoplus_{i = 1}^r\bigoplus_{j = 1}^{n_i}\frac{\mathbb{k}[X]}{(f_i^{s_{ij}})} $$
with each $V[f_i]$ the $f_i$-torsion of $V$.
The general form is rather $$\bigoplus_{i=1}^r\bigoplus_{j=1}^{a_i} \left(\frac{k[X]}{(f_i^{j})}\right)^{n_{i,j}}$$ where $m_T=f_1^{a_1}\cdots f_r^{a_r}$ with the $f_i$ distinct monic irreducibles. Here the $n_{i,j}$ are non-negative integers, and $n_{i,a_i}>0$ in order to ensure that $m_T$ really is the minimum polynomial of $T$.