Algebraic and topological aspects of quasi-prime ideals

Subject Areas : Commutative algebra

M.  Aghajani ¹ (Department of Mathematics, Faculty of Basic Sciences, P. O. Box 55136-553, University of Maragheh, Maragheh, Iran)
A.  Tarizadeh ² (Department of Mathematics, Faculty of Basic Sciences, P. O. Box 55136-553, University of Maragheh, Maragheh, Iran)

Received: 2020-06-29 Accepted : 2020-09-24 Published : 2020-09-01

Keywords: Quasi-prime ideal, connected component, t-functor,

Abstract :

In this paper, we define the new notion of quasi-prime ideal which generalizesat once both prime ideal and primary ideal notions. Then a naturaltopology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basicproperties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is proved that ifthe Grothendieck t-functor is applied on the quasi-prime spectrumthen the prime spectrum is deduced. It is also shown that there arethe cases that the prime spectrum and quasi-prime spectrum do notbehave similarly. In particular, natural topological spaces withoutclosed points are obtained.

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