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The Coproduct of Ideals and the Coprime Spectrum
Abstract
We introduce the coproduct of ideals and the notion of coprime ideal, extending the Heyting-algebra perspective on ideal lattices to arbitrary commutative rings. The resulting coprime spectrum is a topological space, defines a covariant functor on CF-morphisms of commutative rings, i.e. morphisms that extend ideals and are coprime faithful, classifies fields among integral domains, and in dual rings is homeomorphic to the prime spectrum. This offers an elementary, lattice-theoretic counterpart to the geometric study of nilpotents.
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BibTeX
@misc{murphyhernandez2026coproduct,
title = {{The Coproduct of Ideals and the Coprime Spectrum}},
author = {Frank Murphy-Hernandez and Eduardo Leon-Rodriguez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23351},
eprint = {2609.23351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23351},
}
RIS
TY - UNPB
TI - The Coproduct of Ideals and the Coprime Spectrum
AU - Frank Murphy-Hernandez
AU - Eduardo Leon-Rodriguez
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.23351
UR - https://arxiv.org/abs/2609.23351
AB - We introduce the coproduct of ideals and the notion of coprime ideal, extending the Heyting-algebra perspective on ideal lattices to arbitrary commutative rings. The resulting coprime spectrum is a topological space, defines a covariant functor on CF-morphisms of commutative rings, i.e. morphisms that extend ideals and are coprime faithful, classifies fields among integral domains, and in dual rings is homeomorphic to the prime spectrum. This offers an elementary, lattice-theoretic counterpart to the geometric study of nilpotents.
ER -