Let $p,q \in\mathbb{C}[x,y]-\mathbb{C}$, with $p \neq q$. Let $I=\langle p,q \rangle$, namely, $I$ is the ideal of $\mathbb{C}[x,y]$ generated by $p$ and $q$.
Example: We can write $p=Ap+Bq$, with $A=1-q$ and $B=p$; indeed, $p=(1-q)p+pq$.
Is it possible to find a general form of $A$ and $B$ satisfying $p=Ap+Bq$?
Is it true that necessarily $A,B \in \mathbb{C}[p,q]$, as in the example?
Any hints and comments are welcome!
Let $A,B$ such that $Ap+Bq=p$, so that $(A-1)p=-Bq$.
Let $d$ be a gcd of $p$ and $q$, and write $p=rd$ and $q=sd$, where $r,s$ are coprime.
Then $(A-1)rd=-Bsd$, that is $(A-1)r=-Bs$. Since $r,s$ are coprime, $A-1=Cs$ for some $C$, and we deduce that $B=-Cr$.
All in all, $A=1+C(q/d)$ and $B=-C(p/d)$, where $C$ is arbitrary and $d$ is a gcd of $p$ and $q$. In particular, the answer in no for your second question.