Prove that $\eta: R/(a)\to R/(p_1^{\alpha_1})\oplus \cdots \oplus R/(p_r^{\alpha_r})$ is an epimorphism

Question

Let $R$ be PID, and $a \in R$ with $a=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ its prime factorization. I'd like a Hint in proving that $$ x+(a) \to \eta(x+(a))=(x+(p_1^{\alpha_1}),\ldots,x+(p_r^{\alpha_r})) $$ is an epimorphism. My approach is too faint, so I didn't write it.