Support of localization of a module at a minimal prime over the support of the original module

Question

If $M$ is a non-zero module over a commutative ring $R$ (not necessarily Noetherian), and $P$ is a minimal prime in $\mathrm{Supp}(M)$, then is it true that $\mathrm{Supp}(M_P)=\{PR_P\}$ (where $M_P$ is considered as $R_P$-module) ?

[The definition of support is: $\mathrm{Supp}(M):=\{Q \in \mathrm{Spec}(R): M_Q \ne 0\}$]

Answer 1

Yes. $Spec(R_p)=\{q_p\mid q \in SpecR, q\subset p\}$.If $q\subset p$ is prime, then $(M _p)_{q_p}\cong M _q$ is not 0 iff $q$ lies in $SuppM$. The condition is $p$ is minimal in $SuppM$.