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I'm trying to solve the following problem: let $p$ be a prime in a principal ideal domain $R$. Then if $M$ is an $R$-module, I have already shown that $(p^k)M/(p^{k+1})M$ is a module over the field $R/(p)$. Now in the case where

$$M = R^n \oplus R/(p^{s_1}) \oplus \ldots \oplus R/(p^{s_r}),$$

I am trying to compute the dimension of $(p^k)M/(p^{k+1})M$ as a vector space over the field $R/(p)$. But I am not sure of the best way to proceed. Any hints or solutions would be appreciated!

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    Your first step makes absolutely no sense. You cannot use Chinese remainder theorem if the ideals are not co-prime. – 2017-02-01
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    Oops, my mistake. Thanks. – 2017-02-01

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