Abstract
A ring is called a zip ring (Carl Faith coined this term) if every faithful ideal contains a finitely generated faithful ideal. By first proving that a reduced ring is a zip ring if and only if every dense element of the frame of its radical ideals is above a compact dense element, we study algebraic frames with the property stated in the title. We call them zipped. They generalize the coherent frames of radical ideals of zip rings, but (unlike coherent frames) they need not be compact. The class of zipped algebraic frames is closed under finite products, but not under infinite products. If the coproduct of two algebraic frames is zipped, then each cofactor is zipped. If the ring is not necessarily reduced, then its frame of radical ideals is zipped precisely when the ring satisfies what in the literature is called the weak zip property. For a Tychonoff space X, we show that C(X) is a zip ring if and only if X is a finite discrete space.
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References
Azarpanah, F.: Intersection of essential ideals in \(C(X)\). Proc. Am. Math. Soc. 125, 2149–2154 (1997)
Banaschewski, B.: Radical ideals and coherent frames. Comment. Math. Univ. Carolin. 37, 349–370 (1996)
Banaschewski, B.: Gelfand and exchange rings: their spectra in pointfree topology. Arab. J. Sci. Eng. 25, 3–22 (2003)
Banaschewski, B., Pultr, A.: Booleanization. Cah. Topol. Géom. Différ. Catég. 37, 41–60 (1996)
Bhattacharjee, P., Dube, T.: On fraction-dense algebraic frames. Algebra Universalis 83, 6 (2022)
Faith, C.: Rings with ascending condition on annihilators. Nagoya Math. J. 27, 179–191 (1966)
Faith, C.: Rings with zero intersection property: zip rings. Publ. Mat. 33, 329–338 (1989)
Faith, C.: Annihilator ideals, associated primes and Kasch–McCoy commutative rings. Commun. Algebra 19, 1867–1892 (1991)
Hager, A.W., Martínez, J.: Patch-generated frames and projectable hulls. Appl. Categ. Struct. 15, 49–80 (2007)
Ighedo, O., McGovern, W.W.: On the lattice of \(z\)-ideals of a commutative ring. Topol. Appl. 273, 106969 (2020)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Martínez, J.: Unit and kernel systems in algebraic frames. Algebra Universalis 55, 13–43 (2006)
Martínez, J.: An innocent theorem of Banaschewski applied to an unsuspecting theorem of De Marco, and the aftermath thereof. Forum Math. 25, 565–596 (2013)
Martínez, J., Zenk, E.R.: When an algebraic frame is regular. Algebra Universalis 50, 231–257 (2003)
Martínez, J., Zenk, E.R.: Epicompletion in frames with skeletal maps, II: Compact normal joinfit frames. Appl. Categ. Struct. 17, 467–486 (2009)
Ouyang, L.: Ore extensions of zip rings. Glasgow Math. J. 51, 525–537 (2009)
Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Springer, Basel (2012)
Zelmanowitz, J.M.: The finite intersection property on annihilator right ideals. Proc. Am. Math. Soc. 57, 213–216 (1976)
Acknowledgements
We are most grateful to the referee for comments and suggestions that have improved the first version of this paper. The first-named author acknowledges funding from the National Research Foundation of South Africa under Grant 129256.
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Presented by W. Wm. McGovern.
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Dube, T., Blose, S. Algebraic frames in which dense elements are above dense compact elements. Algebra Univers. 84, 3 (2023). https://doi.org/10.1007/s00012-022-00799-w
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DOI: https://doi.org/10.1007/s00012-022-00799-w