Zero sets in pointfree topology and strongly $z$-ideals

Document Type: Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences‎, ‎Hakim Sabzevari University‎, ‎Sabzevar‎, ‎Iran.

2 Department of Mathematics‎, ‎Gorgan Branch‎, ‎Islamic Azad University‎, ‎Gorgan‎, ‎Iran.

Abstract

In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L), which is ring of real-valued continuous functions for frame L, does not have more than one element. Also, z-filters are introduced in terms of pointfree topology. Then the relationship between z-filters and ideals, particularly maximal ideals, is examined. Finally, it is shown that the family of all zero sets is a base for the collection of closed sets.

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