I have a question regarding the quotient of a infinite product of groups. Suppose $(G_{i})_{i \in I}$ are abelian groups with $|I|$ infinite and each $G_i$ has a normal subgroup $N_i$. Is it true in general that $\prod_{i \in I} G_i/ \prod_{i \in I} N_i \cong \prod_{i\in I} G_i/N_i$ More specifically, is it true that $\prod_{p_i \text{prime}} \mathbb{Z}_{p_i} / \prod_{p_i \text{prime}} p^{e_i}\mathbb{Z}_{p_{i}} \cong \prod_{p_i \text{prime},e_{i} \leq \infty} \mathbb{Z}/p_{i}^{e_i}\mathbb{Z} \times \prod_{p_i \text{prime},e_{i} = \infty}\mathbb{Z}_{p_i}$ where $\mathbb{Z}_{p_i}$ stands for the $p_i$-adic integers and $p_i^\infty \mathbb{Z}_{p_i}=0$ and all $e_i$ belong to $\mathbb{N} \cup \{\infty\}$.
Any help would be appreciated.