Two generalizations of LCM-stable extensions

Let R ⊆ T be an extension of integral domains, X be an indeterminate over T, and R[X] and T [X] be polynomial rings. Then R ⊆ T is said to be LCM-stable if (aR∩bR)T = aT∩bT for all 0 ≠ a, b ∈ R. Let w_A be the so-called w-operation on an integral domain A. In this paper, we introduce the notions of w(e)- and w-LCM stable extensions: (i) R ⊆ T is w(e)-LCM-stable if ((aR ∩bR)T)w_T = aT ∩bT for all 0 ≠ a, b ∈ R and (ii) R ⊆ T is w-LCM-stable if ((aR ∩ bR)T)w_R = (aT ∩ bT)w_R for all 0≠a,b∈R. We prove that LCM-stable extensions are both w(e)-LCM- stable and w-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., PvMD), then R ⊆ T is w(e)-LCM-stable (resp., w-LCM-stable) if and only if R[X] ⊆ T [X] is w(e)-LCM-stable (resp., w-LCM-stable).