A closer look at primal and pseudo-irreducible ideals with applications to rings of functions

Abstract

We return to the work of Banaschewski and extract from it a theorem of Fuchs, Heinzer, and Olberding. As an application of Fuchs-Heinzer-Olberding’s theorem, we generalize a result of Gillman and Kohls. We study pseudo-irreducible ideals and show that every ideal of a pm-ring is the (not necessarily finite) intersection of pairwise comaximal pseudo-irreducible ideals. After some general results, the article focuses on primal and pseudo-irreducible ideals in rings of continuous functions. We determine when every pseudo-irreducible ideal of C(X) is primal. We give a characterization of spaces X for which every O^p is a primal ideal of C(X).

KEYWORDS:

• Irreducible ideal
• primal ideal
• pseudo-irreducible ideal
• pseudoprime ideal
• ring of continuous functions

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13C05
• 13F99
• 54C40

Acknowledgments

The authors would like to thank the anonymous referee for reading this article carefully and giving valuable comments, which have improved the quality of this manuscript.

Additional information

Funding

Zahra Keshtkar was supported by a grant from Basic Sciences Research Fund (No. BSRF-math-399-07).