Commun. Korean Math. Soc. 2024; 39(1): 259-266
Online first article January 24, 2024 Printed January 31, 2024
https://doi.org/10.4134/CKMS.c230095
Copyright © The Korean Mathematical Society.
Amartya Goswami
National Institute for Theoretical and Computational Sciences (NITheCS)
The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to $r$-strongly irreducible $r$-ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are $T_0$, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize $r$-arithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of $r$-maximal $r$-ideals and $r$-prime $r$-ideals to be dense in the corresponding terminal spaces.
Keywords: $r$-strongly irreducible $r$-ideals, $r$-arithmetic monoids, Zariski topology, generic points
MSC numbers: Primary 20M12, 20M14, 54F65
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