Characterizations of U-frames and frames that are finitely a U-frame

In this article, we give algebraic characterizations of U-frames in terms of ring-theoretic properties of the ring RL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}L$$\end{document} of real-valued continuous functions on a completely regular frame L. We show that a frame is a U-frame if and only if it is an F-frame and its & Ccaron;ech-Stone compactification is zero-dimensional. We will also introduce frames that are finitely a U-frame and we will characterize them in terms of ring-theoretic properties in RL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}L$$\end{document}.