Let $R$ be an integral domain with fraction field $K$ such that for every ring $R \subseteq S \subseteq K$ , the ring extension $R \subseteq S$ satisfies Going Down property i.e. for any chain of prime ideals $P_1 \subseteq P_2$ in $R$ and any prime ideal $Q_2$ in $S$ with $Q_2 \cap R=P_2$, there is a prime ideal $Q_1$ of $S$ such that $Q_1 \cap R=P_1$ .

My question is : If $0\ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is principal for every $b\in R$ ?