On purely-prime ideals with applications

Abstract

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.

Keywords:

• Gelfand ring
• mp-ring
• pure ideal
• purely-maximal ideal
• purely-prime ideal
• semi-Noetherian ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 14A05
• 13C11
• 54B35
• 54B40

Acknowledgements

We would like to thank the referee for very careful reading of the paper and for his/her valuable suggestions and comments which improved the paper.