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 Op is a primal ideal of C(X).
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.