On purely-prime ideals with applications

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.