Localic remote points revisited

We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension XL! L determined by a set X of filters in L, we show that if there is an ultrafilter in X then the extension has a remote point. In particular, if a completely regular frame L has a maximal completely regular filter which is an ultrafilter, then L! L has a remote point, where L is the Stone- ˇ Cech compactification of L. We prove that in certain extensions associated with radical ideals and '-ideals of reduced f -rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible '-ideals. Concerning coproducts, we show that if M1 ! L1 and M2 ! L2 are extensions of T1-frames, then each of these extensions has a remote point if the extension M1 M2! L1 L2 has a remote point.