Length of module over finite local ring
Let $R$ be a finite local principal ideal ring (think $\mathbb{Z}$ modulo prime power) with prime ideal $m$ and nilpotency index $q$, $D$ a finite length $R$-module, and consider $D/mD$ (which is isomorphic as an $R$-module to $(R/m) \otimes_R D)$ Suppose $\operatorname{length}_R(D/mD)=a$, then what conditions do we need to force on $D$ so that $\operatorname{length}_R(D)=\operatorname{length}_R(R) \cdot a$?
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When you say need, are you asking for necessary or sufficient conditions?
Atm I'm mainly looking for sufficient conditions, but if you can give me some necessary as well, that would be nice