Let us given a module homomorphism $h: M \to N$, where $M,N$ are $\mathbb Z_p$-module and $\mathbb Z_p$, the ring of $p$-adic integers.
Assume that $\{u,v\}$ is a $\mathbb Z_p$-basis of the image $h(M)$. What can we get by utilizing this information ?
I want to know what would be the possible application once we know that $\{u,v\}$ of the image $h(M)$. Here $h(M)$ is a $\mathbb Z_p$-module of rank $2$.
The value $h(m)$ for $m \in M$ is used in many important formulas in number theory. So because $\{u,v\}$ is basis of $h(M)$, we have $$h(m)= au+bv,$$ for some scalars $a,b \in \mathbb Z_p$. But still we don't know $a,b$. Is it practically possible to compute specific scalars $a,b$ here ?
Suppose $M$ is a free $\mathbb Z_p$ with basis $\{x,y\}$, then $m=cx+dy$ for some scalars $c,d \in \mathbb Z_p$. Therefore, $$h(m)=h(cx+dy)=ch(x)+dh(y).$$
Can someone highlight in other situations where the "knowing the basis of the image of a module under the module homomorphism is important" ?
In the general case, when $M,N$ are $R$-modules being $R$ a ring, if $u,v$ is a basis of $h(M)$ then there exist two elements, $x,y\in M$ such that $h(x)=u$ and $h(y)=v$.
Now, as you has said, you can compute $h$ for any element of the free submodule of $M$ $$ L=Rx\oplus Ry $$
Now, what you have to wonder is: $$ M=L\oplus Ker(h) ? $$
Because in that case, you can consider that your map $h$ is "factored by a projection": given an element $m=\alpha x+\beta y+d$, you send $m$ to $\alpha x+\beta y$ and then apply $h$.
I don't know if this applies in your specific setup, but maybe it gives you a clue to continue searching.