On Lattices of z-ideals of Function Rings

Authors

Themba Dube, University of South Africa
Oghenetega Ighedo, Chapman UniversityFollow

Document Type

Article

Publication Date

3-31-2018

Abstract

An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann²(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡L, respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of L (and, in some cases, the algebraic properties of the ring 𝓡L), those L for which Z(L) and D(L) satisfy the various regularity conditions on algebraic frames introduced by Martínez and Zenk [20]. Every frame homomorphism h : L → M induces a coherent map Z(h) : Z(L) → Z(M). Conditions are given of when this map is closed, or weakly closed in the sense Martínez [19]. The case of openness of this map was discussed in [11]. We also prove that, as in the case of the ring C(X), the sum of two z-ideals of 𝓡L is a z-ideal.

Comments

This article was originally published in Mathematica Slovaca, volume 68, issue 2, in 2018. https://doi.org/10.1515/ms-2017-0099

Recommended Citation

T. Dube & O. Ighedo: On lattices of z-ideals of function rings. Mathematica Slovaca, Vol. 68, 2 (2018), 271 – 284. https://doi.org/10.1515/ms-2017-0099

Peer Reviewed

1

Copyright

De Gruyter