The strong ultrafilter topology on spaces of ideals

The patch/constructible refinement of the Zariski topology on the prime spectrum of a commutative ring is well known and well studied. Recently, Fontana and Loper gave an equivalent definition of this topology using ultrafilters. In this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring. We study various properties of these topologies. As applications we use the ultrafilters to classify all the compact subsets of a spectral space in the Zariski topology and we classify Grothendieck's retrocompact spaces again using ultrafilters.