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.

**Keywords**

**Main Subjects**

[1] M. Aghajani, A. Tarizadeh, Characterizations of Gelfand rings specially clean rings and their dual rings, Results Math 75, 2020:125.

[2] A. J. de Jong et al., Stacks Project, see http://stacks.math.columbia.edu.

[3] C. A. Finocchiaro, M. Fontana, D. Spirito, A topological version of Hilbert’s Nullstellensatz, J. Algebra 461 (2016), 25-41.

[4] L. Fuchs, On quasi-primary ideals, Acta Sci. Math. 11 (3) (1947), 174-183.

[5] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer Math. Soc. 142 (1969), 43-60.

[6] K. Schwede, Gluing schemes and a scheme without closed points, Recent progress in arithmetic and algebraic geometry, Contemp. Math. 386 (2005), 157-172.

[7] A. Tarizadeh, Flat topology and its dual aspects, Comm. Algebra 47 (1) (2019), 195-205.

[8] A. Tarizadeh, Zariski compactness of minimal spectrum and flat compactness of maximal spectrum, J. Algebra Appl. 18, 2019:1950202.