Document Type : Research Paper

**Authors**

Department of Mathematics, Faculty of Basic Sciences, P. O. Box 55136-553, University of Maragheh, Maragheh, Iran

**Abstract**

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is proved that if the Grothendieck t-functor is applied on the quasi-prime spectrum then the prime spectrum is deduced. It is also shown that there are the cases that the prime spectrum and quasi-prime spectrum do not behave similarly. In particular, natural topological spaces without closed points are obtained.

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