Let $R$ be a commutative ring, $\{M_i\}_{i\in I}$ a family of $R$-modules and $\{S_i\subseteq M_i\}_{i\in I}$ be a family of $R$-submodules. Does the following hold true $$\left(\prod_iM_i\right)/\left(\prod_iS_i\right)\cong \prod_{i\in I}\left(M_i/S_i\right)?$$ Where $\prod_{i\in I}M_i$ is a direct product of modules. Is there a documented Reference in a Text Book/or Published Paper for this isomorphism.
I have a reference for this isomorphism for direct summands in (Advanced Modern Algebra, Part 1 by Joseph J. Rotman (Proposition B-2.20)). That is, $$\left(\bigoplus_iM_i\right)/\left(\bigoplus_iS_i\right)\cong \bigoplus_{i\in I}\left(M_i/S_i\right).$$ However, I don't have any reference for direct products other than (Modifying $\frac{\prod_\alpha A_\alpha}{\prod_\alpha B_\alpha}\simeq \prod_\alpha\frac{A_\alpha}{B_\alpha}$ for direct sums)!