Strongly irreducible ring and its S spectrum

Hemin A.  Ahmad; Parween A.  Hummadi

doi:10.24996/ijs.2024.65.7.30

Authors

Hemin A. Ahmad Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq https://orcid.org/0000-0002-3095-7781
Parween A. Hummadi Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq

DOI:

https://doi.org/10.24996/ijs.2024.65.7.30

Keywords:

Strongly irreducible ideal, Strongly irreducible ring, S spectrum, S.spec(S) topology, S Zariski topology

Abstract

Let Κ be a proper ideal of a commutative ring S. Then Κ is a strongly irreducible (SI) ideal if for any two ideals C and D of S, C∩D⊆ Κ implies C ⊆ Κ or D ⊆ Κ. We say a ring is strongly irreducible (SI) if all its ideals are strongly irreducible. In this paper some properties of such ring are given. The relations between SI rings and some types of rings are also studied. Furthermore, for an SI ring S we study the S spectrum of S and the S.spec(S) topology.