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PreprintCC BY 4.0arXiv

Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals

  • Reza Nikandish
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Abstract

We describe the ideals of an idealization Dltimes M by triples (I,N,φ), where I is an ideal of D, N is a submodule of M with IM⊆ N, and φ∈HomD(I,M/N). For a domain D and torsion-free M, the nonzero proper ideals of Dltimes M with nonzero annihilator are exactly 0ltimes N. As an application, for R=ℝ[[t]]ltimesℝ[[t]], the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality 2ℵ0. This answers negatively a question of Behboodi and Rakeei.

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Cite

BibTeXRIS
BibTeX
@misc{nikandish2026ideal,
  title = {{Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals}},
  author = {Reza Nikandish},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25157},
  eprint = {2609.25157},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25157},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals
AU  - Reza Nikandish
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25157
UR  - https://arxiv.org/abs/2609.25157
AB  - We describe the ideals of an idealization Dltimes M by triples (I,N,φ), where I is an ideal of D, N is a submodule of M with IM⊆ N, and φ∈Hom_D(I,M/N). For a domain D and torsion-free M, the nonzero proper ideals of Dltimes M with nonzero annihilator are exactly 0ltimes N. As an application, for R=ℝ[[t]]ltimesℝ[[t]], the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality 2^ℵ_0. This answers negatively a question of Behboodi and Rakeei.
ER  -