This exercise started asking to prove that there's a morphism between $R/(gh) \rightarrow R/(g) \times R/(h)$ and I proved that the canonical projection is in fact well defined there. Then it asked to find it's kernel, which I identified as the ideal generated by the least common multiple of g and h.
So I was able to see that the kernel of the projection map between $\mathbb{C}[x]$ and $\prod_{i=1}^n \mathbb{C}[x]/(X-z_i)^{r_i}$ is indeed $f$. But now I have to see that it's actually an epimorphism and I'm having quite a bit of trouble there, and I'm not so sure it actually is.
Let's say I have $(\bar{p}_1,...,\bar{p}_n) \in \prod_{i=1}^n \mathbb{C}[x]/(X-z_i)^{r_i}$. I then should have a polynomial $P(x) = p_1(x) + (X-z_1)^{r_1}g_1(x) =\cdots= p_n(x) + (X-z_n)^{r_n}g_n(x)$, but I'm not sure how I would be able to construct it.
Any help would be greatly appreciated.