Concerning the frame of minimal prime ideals of pointfree function rings

Document Type : Research Paper

Author

Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa.

Abstract

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of continuous real-valued functions on $L$. We study the frame $mathfrak{O}(Min(mathcal{R}L))$ of minimal prime ideals of $mathcal{R}L$ in relation to $beta L$. For $Iinbeta L$, denote by $textit{textbf{O}}^I$ the ideal ${alphainmathcal{R}Lmidcozalphain I}$ of $mathcal{R}L$. We show that sending $I$ to the set of minimal prime ideals not containing $textit{textbf{O}}^I$ produces a $*$-dense one-one frame homomorphism $beta Ltomathfrak{O}(Min(mathcal{R}L))$ which is an isomorphism if and only if $L$ is basically disconnected.

Keywords


[1] R.N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Dissertationes Mathematicae (Rozprawy Mat.), Vol. 412 (2002), 62pp.
[2] B. Banaschewski, The real numbers in pointfree topology, Textos de Matematica Serie B, No. 12, Departamento de Matematica da Universidade de Coimbra, 1997.
[3] B. Banaschewski, C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 577-587.
[4] B. Banaschewski and M. Sioen, Ring ideals and the Stone-  Cech compacti cation in pointfree topology, J. Pure Appl. Algebra 214 (2010), 2159-2164.
[5] T. Dube, Some ring-theoretic properties of almost P-frames, Alg. Univ., 60 (2009), 145-162.
[6] T. Dube, Contracting the socle in rings of continuous functions, Rend. Sem. Mat. Univ. Padova 123 (2010), 37-53.
[7] T. Dube, Notes on pointfree disconnectivity with a ring-theoretic slant, Appl. Categor. Struct. 18 (2010), 55-72.
[8] T. Dube and M. Matlabyane, Concerning some variants of C-embedding in pointfree topology, Top. Appl. 158 (2011), 2307-2321.
[9] T. Dube and I. Naidoo, On openness and surjectivity of lifted frame homomorphisms, Top. Appl. 157 (2010), 2159-2171.
[10] G. Gruenhage, Products of cozero complemented spaces, Houst. J. Math. 32 (2006), 757-773.
[11] A.W. Hager and J. Martnez, Fraction-dense algebras and spaces, Canad. J. Math. 45 (1993), 977-996.
[12] A.W. Hager and J. Martnez, Patch-generated frames and projectable hulls, Appl. Categor. Struct. 15 (2007), 49-80.
[13] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110-130.
[14] J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel (2012).
[15] S. Woodward, On f-rings which are rich in idempotents, PhD thesis (1992), University of Florida.