vector space constructed through a torsion module

Question

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form: $$ M = R/(p^{e_1}) \oplus \dots \oplus R/(p^{e_t}). $$

Let now be $_pM = \{m \in M \mid p m = 0\}$.

I now want to show that $_pM$ is a $R/(p)$-vector space of dimension $t$.

Thanks in advance. I'm not very used to these constructions.

Answer 1

$${}_pM=\bigoplus_{i=1}^t(p^{e_i-1})/(p^{e_i})\simeq\bigoplus_{i=1}^tR/(p).$$

Indeed, if $\bar x=x+(p^{e_i})$ is an element of $R/(p^{e_i})$, we have: $$p \bar x=\bar 0\iff px\in(p^{e_i})\iff x\in (p^{e_i-1})$$ since $p$ is a prime element.