We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s ◦ f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally cl...

In [1], Afkhami and Khashyarmanesh introduced the cozero-divisor graph of a ring, Γ′(R), which examines relationships between principal ideals. We continue investigating the algebraic implications of the graph by developing the reduced cozero-divisor graph, which is a simpler analog. Acknowledgements: This research was undertaken at the Wabash College Mathematics REU in Crawfordsville, Indiana....

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a σ-frame and to Alexandroff spaces.

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

For a zero-dimensional topological space X and totally ordered field F with interval topology, Cc(X, F) denotes the ring consisting of field-valued continuous functions countable range on X. This article aims to study investigate rings quotients Cc(X,F). Qc(X,F) (resp. qc(X,F)), maximal classical) Cc(X,F) as modified analogue Q(X) q(X)), C(X) are characterized. It is proved that Qc(S), subring ...

A bijective correspondence between strong inclusions and compactifications in the setting of σ-frames is presented. The category of uniform σ-frames is defined and a description of the Samuel compactification is given. It is shown that the Samuel compactification of a uniform frame is completely determined by the σ-frame consisting of its uniform cozero part, and consequently, any compactificat...