Quasi-prime ideals

TL;DR

This paper introduces the concept of quasi-prime ideals, generalizing prime and primary ideals, and explores their properties and topological structure, revealing differences from prime spectra in certain cases.

Contribution

It defines quasi-prime ideals, develops a topology on their spectrum, and connects this to the prime spectrum via the Grothendieck t-functor, highlighting new structural insights.

Findings

Quasi-prime spectrum generalizes prime spectrum.

A topology on quasi-prime spectrum extends Zariski topology.

Prime and quasi-prime spectra can behave differently.

Abstract

In this paper, the new concept of quasi-prime ideal is introduced which at the same time generalizes the `prime ideal' and `primary ideal' notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which generalizes the Zariski 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.