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Increasing sequences of principal left ideals of $\beta \mathbb{Z}$ are finite

Volume 258 / 2022

Yevhen Zelenyuk Fundamenta Mathematicae 258 (2022), 225-235 MSC: Primary 22A15, 54H20; Secondary 05D10, 54D80. DOI: 10.4064/fm17-8-2021 Published online: 10 January 2022


We show that increasing sequences of principal left ideals of $\beta \mathbb {Z}$ are finite. As a consequence, $\beta \mathbb {Z}\setminus \mathbb {Z}$ is a disjoint union of maximal principal left ideals of $\beta \mathbb {Z}$. Another consequence is that increasing chains of idempotents ($p\le q\Leftrightarrow p+q=q+p=p$) in $\beta \mathbb {Z}$ are finite. All these are answers to long-standing open questions.


  • Yevhen ZelenyukSchool of Mathematics
    University of the Witwatersrand
    Private Bag 3, Wits 2050
    Johannesburg, South Africa

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