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Classes of Commutative Clean Rings

Wolf IberkleidWarren Wm. McGovern — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

Let A be a commutative ring with identity and I an ideal of A . A is said to be I - c l e a n if for every element a A there is an idempotent e = e 2 A such that a - e is a unit and a e belongs to I . A filter of ideals, say , of A is if for each I there is a finitely generated ideal J such that J I . We characterize I -clean rings for the ideals 0 , n ( A ) , J ( A ) , 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.

C * -points vs P -points and P -points

Jorge MartinezWarren Wm. McGovern — 2022

Commentationes Mathematicae Universitatis Carolinae

In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 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....

When Min ( G ) - 1 has a clopen π -base

Ramiro Lafuente-RodriguezWarren Wm. McGovern — 2021

Mathematica Bohemica

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...

The regular topology on C ( X )

Wolf IberkleidRamiro Lafuente-RodriguezWarren Wm. McGovern — 2011

Commentationes Mathematicae Universitatis Carolinae

Hewitt [Rings of real-valued continuous functions. I., Trans. Amer. Math. Soc. 64 (1948), 45–99] defined the m -topology on C ( X ) , denoted C m ( X ) , and demonstrated that certain topological properties of X could be characterized by certain topological properties of C m ( X ) . For example, he showed that X is pseudocompact if and only if C m ( X ) 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...

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