Generalized divisor topology of commutative rings
Abstract
Let R be a commutative ring with nonzero identity and let R# denote the set of its nonzero nonunits. We extend the divisor topology D(R), previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology GD(R) on EC(R#). Its basic open sets are Ba={[b]∈ EC(R#): b| an for some n≥ 1}. The relation [b]∈ Ba Longleftrightarrow √(aR)⊆√(bR) shows that GD(R) records radical divisibility among principal ideals. We prove that GD(R) is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the T0 and discrete properties and of the equality GD(R)=D(R). We also determine the isolated points of GD(R). Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain R, GD(R) is compact if and only if R is a G-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of GD(R) with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.
Related
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BibTeX
@misc{koc2026generalized,
title = {{Generalized divisor topology of commutative rings}},
author = {Suat Koç and İrem Doğan and Dilara Erdemir and Ünsal Tekir},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14867},
eprint = {2609.14867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14867},
}
RIS
TY - UNPB
TI - Generalized divisor topology of commutative rings
AU - Suat Koç
AU - İrem Doğan
AU - Dilara Erdemir
AU - Ünsal Tekir
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14867
UR - https://arxiv.org/abs/2609.14867
AB - Let R be a commutative ring with nonzero identity and let R^# denote the set of its nonzero nonunits. We extend the divisor topology D(R), previously studied for integral domains, to arbitrary commutative rings and introduce the generalized divisor topology GD(R) on EC(R^#). Its basic open sets are B_a={[b]∈ EC(R^#): b| a^n for some n≥ 1}. The relation [b]∈ B_a Longleftrightarrow √(aR)⊆√(bR) shows that GD(R) records radical divisibility among principal ideals. We prove that GD(R) is an Alexandrov space and identify its Kolmogorov quotient with the poset of radicals of nonzero proper principal ideals. This description yields characterizations of the T_0 and discrete properties and of the equality GD(R)=D(R). We also determine the isolated points of GD(R). Further, we characterize nestedness, compactness, the Lindelöf property, and Noetherianity in terms of the order structure of radicals of principal ideals. In particular, for an integral domain R, GD(R) is compact if and only if R is a G-domain, while for a UFD the Lindelöf and Noetherian properties are determined by the number of nonassociate prime elements. Finally, we study the interaction of GD(R) with multiplication and describe the behavior of its Kolmogorov quotient under surjective homomorphisms with nil kernel.
ER -