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Let $A$ be a commutative ring with identity and $I$ an ideal of $A$. $A$ is said to be $I$-$clean$ if for every element $a\in A$ there is an idempotent $e=e^2\in A$ such that $a-e$ is a unit and $ae$ belongs to $I$. A filter of ideals, say $ℱ$, of $A$ is if for each $I\in ℱ$ there is a finitely generated ideal $J\in ℱ$ such that $J\subseteq I$. We characterize $I$-clean rings for the ideals $0$, $n\left(A\right)$, $J\left(A\right)$, and $A$, in terms of the frame of multiplicative Noetherian filters of ideals of $A$, as well as in terms of more classical ring properties.

Let $C(X,\mathbb{Z})$, $C(X,\mathbb{Q})$ and $C\left(X\right)$ denote the $\ell$-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space $X$, respectively. Characterizations are given for the extensions $C(X,\mathbb{Z})\le C(X,\mathbb{Q})\le C\left(X\right)$ to be rigid, major, and dense.

It is our aim to contribute to the flourishing collection of knowledge centered on the space of minimal prime subgroups of a given lattice-ordered group. Specifically, we are interested in the inverse topology. In general, this space is compact and $T_1$, but need not be Hausdorff. In 2006, W. Wm. McGovern showed that this space is a boolean space (i.e. a compact zero-dimensional and Hausdorff space) if and only if the $l$-group in question is weakly complemented. A slightly weaker topological property...

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a $C^{*}$-point....

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the $m$-topology on $C\left(X\right)$, denoted $C_m\left(X\right)$, and demonstrated that certain topological properties of $X$ could be characterized by certain topological properties of $C_m\left(X\right)$. For example, he showed that $X$ is pseudocompact if and only if $C_m\left(X\right)$ is a metrizable space; in this case the $m$-topology is precisely the topology of uniform convergence. What is interesting with regards to the $m$-topology is that it is possible, with...