Journal of the Australian Mathematical Society

Research Article

STRONGLY IRREDUCIBLE IDEALS

A. AZIZIa1

a1 Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71454, Iran (email: aazizi@shirazu.ac.ir, a_azizi@yahoo.com)

Abstract

A proper ideal I of a ring R is said to be strongly irreducible if for each pair of ideals A and B of R, $A\cap B \subseteq I$ implies that either $A \subseteq I$ or $B \subseteq I$. In this paper we study strongly irreducible ideals in different rings. The relations between strongly irreducible ideals of a ring and strongly irreducible ideals of localizations of the ring are also studied. Furthermore, a topology similar to the Zariski topology related to strongly irreducible ideals is introduced. This topology has the Zariski topology defined by prime ideals as one of its subspace topologies.

(Received April 24 2006)

(Accepted August 22 2007)

2000 Mathematics subject classification

  • 13A15;
  • 13C05;
  • 13E05;
  • 13F99

Keywords and phrases

  • absolutely flat rings;
  • arithmetical rings;
  • generalizations of Dedekind domains;
  • Laskerian rings;
  • strongly irreducible ideals;
  • Zariski topology