{
  "abstract": "We describe the ideals of an idealization $D\\ltimes M$ by triples $(I,N,φ)$, where $I$ is an ideal of $D$, $N$ is a submodule of $M$ with $IM\\subseteq N$, and $φ\\in\\operatorname{Hom}_D(I,M/N)$. For a domain $D$ and torsion-free $M$, the nonzero proper ideals of $D\\ltimes M$ with nonzero annihilator are exactly $0\\ltimes N$. As an application, for $R=\\mathbb R[[t]]\\ltimes\\mathbb R[[t]]$, the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality $2^{\\aleph_0}$. This answers negatively a question of Behboodi and Rakeei.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25157",
  "author_total": 1,
  "authors": [
    {
      "name": "Reza Nikandish"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wzav0jkc7gvp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25157",
  "fulltext_url": null,
  "id": "3wzav0jkc7gvp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25157",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214121263",
  "pdf_url": "https://arxiv.org/pdf/2609.25157",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wzav0jkc7gvp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25157",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25157"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals",
  "title_text": "Ideal Structures in Idealizations and Cardinalities of Annihilating Ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/3wzav0jkc7gvp",
  "venue": "arXiv",
  "versions": []
}