Let $R = \mathbb{R}[x,y]$ be bivariate polynomial ring and $I_1 = \langle f_1 \rangle$, $I_2 = \langle f_2 \rangle$ and $I_3 = \langle f_3 \rangle$ be three principal ideals of $R$. Consider the following map.
$\alpha : R^3 \to \dfrac{R}{I_1} \oplus \dfrac{R}{I_2} \oplus \dfrac{R}{I_3}$, with the following rule
$\alpha (a,b,c) = (a - b + I_1 , b - c + I_2 , a- c + I_3)$.
My question is that, how can I check whether $\alpha$ is surjective or not? What I've done is that, let $(\bar{p} , \bar{q} , \bar{r}) \in \dfrac{R}{I_1} \oplus \dfrac{R}{I_2} \oplus \dfrac{R}{I_3}$ where $\bar{p} = p + I_1$, $\bar{q} = q + I_2$ , $\bar{r} = r + I_3$. So can I find such $a,b,c \in R$ that $a - b - p \in I_1$ , $b - c - q \in I_2$ and $a - c - r \in I_3$? Here is where I stuck.
Also how can $f_1 , f_2$ and $f_3$ effect the surjectivity of $\alpha$ ?