In this paper, new algebraic and topological results on purely-prime ideals
of a commutative ring (pure spectrum) are obtained. Particularly,
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.