Integral domain with a quotient field
Given $x\in K$ you have an ideal $I_x\subseteq A$ consisting of elements $b$ such that $bx\in A$. For any $x\in K$ there are two possibilities: $I_x=A$ or $I_x\subseteq M$ for some $M\in {\rm Spm}(A)$.
In the first case $x\in A$, and in the second case $x\notin A_M$. Thus $$\cap_{M\in {\rm Spm(A)}} A_M\subseteq A.$$
The inclusions: $$A\subseteq\cap_{P\in {\rm Spec}(A)} A_P\subseteq\cap_{M\in {\rm Spm}(A)} A_M,$$ follow trivially from $A\subseteq A_P$ for all primes $P$ and ${\rm Spm}(A)\subseteq {\rm Spec}(A)$.