SAMS Subject Classification Number: 7, 29

Thanks to Grothendieck’s algebraic geometry, the Zariski topology on the prime spectrum of a (commutative) ring (with identity) has been studied a considerable amount. On the other hand, although topologies like the patch, inverse, Alexandroff, coarse upper, coarse lower, interval, Scott, Lawson, etc, have mostly been studied in the general context of topology, lattice theory, and commutative algebra, not much has been done on the spectrum of a ring. The purpose of this talk is to give an overview on some of the existing results about these topologies on spectra of rings and then discuss some new ones.