PreprintCC BY 4.0arXiv
The Cuntz semigroup of rings with stable rank one
Abstract
We study the Cuntz semigroup of arbitrary rings with stable rank one. We prove that Cuntz subequivalence between countably generated projective right modules P, Q is equivalent to P being isomorphic to a pure submodule of Q, whilst Cuntz equivalence amounts to isomorphism.~We also prove order-cancellation of finitely generated projective modules (and, more generally, pairs of modules admitting certain complements) from direct sums with countably generated projective modules.
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BibTeX
@misc{ara2026cuntz,
title = {{The Cuntz semigroup of rings with stable rank one}},
author = {Pere Ara and Francesc Perera and Guillem Quingles},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15670},
eprint = {2609.15670},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15670},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Cuntz semigroup of rings with stable rank one
AU - Pere Ara
AU - Francesc Perera
AU - Guillem Quingles
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15670
UR - https://arxiv.org/abs/2609.15670
AB - We study the Cuntz semigroup of arbitrary rings with stable rank one. We prove that Cuntz subequivalence between countably generated projective right modules P, Q is equivalent to P being isomorphic to a pure submodule of Q, whilst Cuntz equivalence amounts to isomorphism.~We also prove order-cancellation of finitely generated projective modules (and, more generally, pairs of modules admitting certain complements) from direct sums with countably generated projective modules.
ER -