On lattices of z-ideals of function rings
-
Themba Dube
and
Oghenetega Ighedo
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.
This work was supported by the National Research Foundation of South Africa, Grant No. 93514.
Acknowledgement
We thank the referee most heartily for comments that have improved the paper. We express a special word of thanks for drawing our attention to references [7] and [8].
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© 2018 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- On the family of functions with closure of graphs in the Mendez ideals
- The predicate completion of a partial information system
- On lattices of z-ideals of function rings
- Compactifications of partial frames via strongly regular ideals
- Lifting components in clean abelian ℓ-groups
- Invariance of nonatomic measures on effect algebras
- On the alexander dual of path ideals of cycle posets
- Generalized multiplicative derivations in 3-prime near rings
- Cleft extensions for quasi-entwining structures
- Groups with positive rank gradient and their actions
- Certain results on q-starlike and q-convex error functions
- Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate
- Positive periodic solutions for singular high-order neutral functional differential equations
- A special class of functional equations
- Matrix mappings and general bounded linear operators on the space bv
- Skew-symmetric operators and reflexivity
- Derivatives of Hadamard type in vector optimization
- Cardinal functions of the hyperspace of convergent sequences
- Suzuki-type of common fixed point theorems in fuzzy metric spaces
- Second hankel determinat for certain analytic functions satisfying subordinate condition
