Local behaviour of a module localized at a prime ideal

Question

Let $R$ be a commutative ring and $p,q$ be two prime ideals of $R$ with $q\subset p$. We know $(R_p)_{qR_p}\cong R_q $ as rings. Let $M$ be an $R$-module. Is it true that $(M_p)_{qR_p}\cong M_q$ as $R_q$-modules?

Answer 1

Given $S^{-1}M \cong S^{-1}R\otimes_R M$ and $(R_p)_{qR_p} \cong R_q$, we have

$$(M_p)_{qR_p} \cong (R_p)_{qR_p} \otimes_{R_p} M_p \cong (R_p)_{qR_p} \otimes_{R_p} R_p \otimes_R M \cong R_q\otimes_R M \cong M_q$$

which shows the desired isomorphism.