Suppose $R$ to be a ring with unity, and we have a module homomorphism $f: A \to B$, where $A,B$ are $R$-modules.
Suppose the image $f(A)$ is free $R$-module of rank $2$ with $R$-basis $\{u,v\}$. We know exactly the values $u,v$.
Suppose we know the value of $f(a_0)$ for some fixed $a_0 \in A$.
Take an arbitrary $a \in A$, then we have $f(a)=\alpha u+\beta v$ for some scalars $\alpha, \beta \in R$.
Let me add more information. The homomorphism $f$ is series map, and we can compute $f(a)$ for any $a \in A$ by putting $a$ in the series representing $f$ but in that case $f(a)$ will have inaccuracy.
Is it possible to derive an algorithm which starts with $f(a_0)$ and at the end find the coordinates $\alpha, \beta$ so that we can find exact or at least more-accurate (degree of precision) value of $f(a)$ ?
The importance of the question is that-if we can find the coordinates $(\alpha, \beta)$, then we can find the value of $f(a)$ for arbitrary point $a \in A$ with better degree of precision.