Prime-Power Multiplicative Modular Monoid Morphisms
Abstract
We undertake a systematic study of multiplicative monoid morphisms between prime-power modular monoids (ℤ/peℤ,·) and (ℤ/qfℤ,·), for arbitrary primes p,q and integers e,f≥ 1. Our approach proceeds in three stages. We first determine the hom-set Hom(Upe,Uqf) of group morphisms between the corresponding unit groups and exhibit explicit isomorphisms describing its group structure. We then identify the subset ℋpe,qf,v of unit group morphisms that extend to full monoid morphisms on (ℤ/peℤ,·), characterizing it via explicit divisibility conditions and describing its algebraic structure. Finally, we give a complete description of HomMon((ℤ/peℤ,·), (ℤ/qfℤ,·)) and establish an explicit monoid isomorphism between this hom-set and a naturally defined monoid built from unit groups of lower-level modular rings.
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BibTeX
@misc{atalaye2026primepower,
title = {{Prime-Power Multiplicative Modular Monoid Morphisms}},
author = {Joseph Atalaye and Liam Baker and Sophie Marques},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02393},
eprint = {2610.02393},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02393},
}
RIS
TY - UNPB
TI - Prime-Power Multiplicative Modular Monoid Morphisms
AU - Joseph Atalaye
AU - Liam Baker
AU - Sophie Marques
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02393
UR - https://arxiv.org/abs/2610.02393
AB - We undertake a systematic study of multiplicative monoid morphisms between prime-power modular monoids (ℤ/p^eℤ,·) and (ℤ/q^fℤ,·), for arbitrary primes p,q and integers e,f≥ 1. Our approach proceeds in three stages. We first determine the hom-set Hom(U_p^e,U_q^f) of group morphisms between the corresponding unit groups and exhibit explicit isomorphisms describing its group structure. We then identify the subset ℋ_p^e,q^f,v of unit group morphisms that extend to full monoid morphisms on (ℤ/p^eℤ,·), characterizing it via explicit divisibility conditions and describing its algebraic structure. Finally, we give a complete description of Hom_Mon((ℤ/p^eℤ,·), (ℤ/q^fℤ,·)) and establish an explicit monoid isomorphism between this hom-set and a naturally defined monoid built from unit groups of lower-level modular rings.
ER -