THE BANASCHEWSKI-SIOEN NUCLEUS ON AN ALGEBRAIC FRAME

Document Type : Original Manuscript

Authors

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria, South Africa.

10.22044/jas.2025.15685.1927

Abstract

In their construction of the Stone-\v{C}ech compactification using only ring ideals (as opposed to $\ell$-ideals), Banaschewski and Sioen define a certain nucleus on the coherent frame $\Rid(A)$ of radical ideals of a commutative ring $A$. In this paper we extend this nucleus to any algebraic frame that has a dense compact element and in which the meet of two compact elements is compact. Of course, not every such algebraic frame is coherent, so the extension is indeed a genuine extension. We then study some properties of this nucleus which are not considered by Banaschewski and Sioen.

Keywords

References

1. B. Banaschewski, Another look at the localic Tychonoff theorem, comment. Math. Univ. Carolin., 29(4) (1988), 647–656.
2. B. Banaschewski, Gelfand and exchange rings: their spectra in pointfree topology, Arab. J. Sci. Eng., 25 (2000), 3–22.
3. B. Banaschewski, Radical ideals and coherent frames, Comment. Math. Univ. Carolin., 37 (1996), 349–370.
4. B. Banaschewski, J. L. Frith and C. R. A. Gilmour, On the congruence lattice of a frame, Pacific J. Math., 130(2) (1987), 209–213.
5. B. Banaschewski and A. Pultr, Booleanization, Cah. Topol. Géom. Différ. Catég., 37 (1996), 41–60.
6. B. Banaschewski and M. Sioen, Ring ideals and the Stone-Čech compactification in pointfree topology, J. Pure Appl. Algebra, 214(12) (2010), 2159–2164.
7. G. Bezhanishvili, C. Carai and P. J. Morandi, Heyting frames and Esakia duality, (2023), arXiv:2302.07913.
8. P. Bhattacharjee, Maximal d-elements of an algebraic frame, Order, 36 (2019), 377–390.
9. P. Bhattacharjee and T. Dube, On fraction-dense algebraic frames, Algebra Universalis, 83 (2022), Article number: 6.
10. T. Dube and S. Blose, Algebraic frames in which dense elements are above dense compact element, Algebra Universalis, 84 (2023), Article number: 3.
11. T. Dube and O. Ighedo, On the torsion ideal of a homomorphism, Topology Appl., Accepted.
12. S. Endo, On semi-hereditary rings, J. Math. Soc. Japan, 13(2) (1960), 109–119.
13. G. Gratzer, Lattice Theory, W. H. Freeman and Company, San Francisco, 1971.
14. A. W. Hager and J. Martínez, Patch-generated frames and projectable hulls, Appl. Categ. Structures, 15 (2007), 49–80.
15. J. A. Huckaba, Commutative rings with zero divisors, Marcel Dekker Inc., 1988.
16. O. Ighedo and W. Wm. McGovern, On the lattice of z-ideals of a commutative rings, Topology Appl., 273 (2020), Article ID: 106969.
17. P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.
18. M. L. Knox and W. Wm. McGovern, Feebly projectable algebraic frames and multiplicative filters of ideals, Appl. Categ. Structures, 15 (2007), 3–17.
19. J. Martínez, An innocent theorem of Banaschewski applied to an unsuspecting theorem of De Marco, and the aftermath thereof, Forum Math., 25 (2013), 565–596.
20. J. Martínez, Archimedean lattices, Algebra Universalis, 3 (1976), 247–260.
21. J. Martínez, Unit and kernel systems in algebraic frames, Algebra Universalis, 5 (2006), 13–43.
22. J. Martínez and E. R. Zenk, Epicompletion in frames with skeletal maps, II: compact normal joinfit frames, Appl. Categ. Structures, 17 (2009), 467–486. 
 23. J. Martínez and E. R. Zenk, Nuclear typing of frames and spatial selectors, Appl. Categ. Structures, 14 (2006), 35–61.
24. J. Martínez and E. R. Zenk, When an algebraic frame is regular, Algebra Universalis, 50 (2003), 231–257.
25. S. Niefield and K. Rosenthal, Componental nuclei, In: Categorical algebra and its Applications, Springer Lecture Notes in Math., 1348 (1988), 299–306.
26. J. Picado and A. Pultr, Frames and Locales: Topology without Points, Frontiers in Mathematics, Springer, Basel, 2012.
27. Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France, 99 (1971), 265–272.
28. H. Simmons, A curious nucleus, J. Pure Appl. Algebra, 214 (2010), 2063–2073 
Journal of Algebraic Systems
Volume 14, Issue 3
June 2026
Pages 569-602

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