Topologies on the torsion-theoretic spectrum of a noncommutative ring

Let R-sp be the collection of all prime torsion theories on the category of left R-modules over an associative ring R. Three topologies - the order topology, the finitary order topology, and the reverse order topology (in the case that R is left noetherian) - are defined on R-sp and each is shown to exhibit some properties of the Zariski topology on the spectrum of a commutative ring. A fourth topology — the Gillman topology - is defined on E-sp when R is left noetherian and is used to characterize the separation of the reverse order topology.