Given an integer matrix $A \in \mathbb{Z}^{m\times m}$. I know one van find the number of elements in the image of $A$ modulo $p^k$ by looking at the Smith Normal Form, i.e. $S = PAQ$ with $P$ en $Q$ invertible over $\mathbb{Z}$.
Suppose $A$ and $A + B$ with $B\in \mathbb{Z}^{m\times m}$ have rank at least $d$ over the prime field of $p$ elements and over the rational/complex numbers (not sure whether that is relevant), then we have (basically by definition) that the number of elements in $\text{Im}_{p} (A+B)$ is at least $p^d$.
I was wondering what can be said about the number of elements of $\text{Im}_{p^k} p^i A + B$ modulo $p^k$ where $i < k$. I was thinking that the number should be at least $p^{k-i}$.
If we would only have $A$, then the Smith Normal Form modulo $p$ would have at least $d$ ones on the diagonal. Hence, considering the Smith Normal Form modulo $p^k$ and multiplying by $p^i$ gives a diagonal matrix with at least $d$ times $p^i$. This implies that the image contains at least $p^{k-i}$ elements.
Can someone give me some ideas how to tackle the case $p^i A + B$, if it is true?