LCM-splitting sets in some ring extensions

Let S S be a saturated multiplicative set of an integral domain D D . Call S S an lcm splitting set if d D S ∩ D dD_{S}\cap D and d D ∩ s D dD\cap sD are principal ideals for every d ∈ D d\in D and s ∈ S s\in S . We show that if R R is an R 2 R_{2} -stable overring of D D (that is, if whenever a , b ∈ D a,b\in D and a D ∩ b D aD\cap bD is principal, it follows that ( a D ∩ b D ) R = a R ∩ b R ) (aD\cap bD)R=aR\cap bR) <