  In this paper, new algebraic and topological results on purely-prime ideals
of a commutative ring (pure spectrum) are obtained. Especially, Grothendieck
type theorem is obtained which states that there is a canonical correspondence
between the idempotents of a ring and the clopens of its pure spectrum. It is
also proved that a given ring is a Gelfand ring iff its maximal spectrum
equipped with the induced Zariski topology is homeomorphic to its pure
spectrum. Then as an application, it is deduced that a ring is zero dimensional
iff its prime spectrum and pure spectrum are isomorphic. Dually, it is shown
that a given ring is a reduced mp-ring iff its minimal spectrum equipped with
the induced flat topology and its pure spectrum are the same. Finally, the new
notion of semi-Noetherian ring is introduced and Cohen type theorem is proved.
