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Characterizations of t-Almost Dedekind Domains via w-Secondary Modules

  • Hwankoo Kim
  • Hyungtae Baek
  • Haleh Hamdi
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Abstract

Let R be an integral domain and let w denote the w-operation on R. A nonzero GV-torsion-free R-module M is called w-secondary if, for every r∈ R, either M/rM is GV-torsion or rnM=0 for some n∈N. A w-module M is said to admit a w-secondary representation if M=(N1+⋯+Nn)w, where each Ni is a w-secondary submodule of M. Such a representation is called minimal if the ideals Pi=√(AnnR(Ni)) are pairwise distinct, and M is called w-representable if it admits such a representation. If the sum can be chosen direct, then M is called strongly w-representable. We characterize w-secondary modules and investigate their localization and w-closure properties. For a nonfield domain R, we prove that R is t-almost Dedekind if and only if wdim R=1 and every w-representable w-module is locally strongly w-representable: each of its nonzero localizations at maximal w-ideals is strongly representable. Global strong w-representability is retained when all the radical annihilators in the representation are nonzero.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026characterizations,
  title = {{Characterizations of t-Almost Dedekind Domains via w-Secondary Modules}},
  author = {Hwankoo Kim and Hyungtae Baek and Haleh Hamdi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16685},
  eprint = {2610.16685},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16685},
}
RIS
TY  - UNPB
TI  - Characterizations of t-Almost Dedekind Domains via w-Secondary Modules
AU  - Hwankoo Kim
AU  - Hyungtae Baek
AU  - Haleh Hamdi
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16685
UR  - https://arxiv.org/abs/2610.16685
AB  - Let R be an integral domain and let w denote the w-operation on R. A nonzero GV-torsion-free R-module M is called w-secondary if, for every r∈ R, either M/rM is GV-torsion or r^nM=0 for some n∈N. A w-module M is said to admit a w-secondary representation if M=(N_1+⋯+N_n)_w, where each N_i is a w-secondary submodule of M. Such a representation is called minimal if the ideals P_i=√(Ann_R(N_i)) are pairwise distinct, and M is called w-representable if it admits such a representation. If the sum can be chosen direct, then M is called strongly w-representable. We characterize w-secondary modules and investigate their localization and w-closure properties. For a nonfield domain R, we prove that R is t-almost Dedekind if and only if wdim R=1 and every w-representable w-module is locally strongly w-representable: each of its nonzero localizations at maximal w-ideals is strongly representable. Global strong w-representability is retained when all the radical annihilators in the representation are nonzero.
ER  -