{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-integral-domain",
  "semantic_family_id": "sf-finite-integral-domain",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限整環"
  },
  "summary": {
    "zh-Hant": "有限交換含一環若1不等於0且沒有非零零因子，則為有限整環。"
  },
  "statement": {
    "zh-Hant": "在同一完整有限交換含一環表上，若對所有非零a,b皆有a乘b不等於0，則為有限整環。完整表可逐對核零因子；每個非零元可逆是利用有限性的後續定理步驟，不偷放入定義。"
  },
  "formula": {
    "tex": "D(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "3d006788565534528bd407df00034466d031a0ee0f9df98fffb1d15d7d9696ed"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "父結構已通過完整有限交換含一環資格。"
    },
    {
      "id": "a2",
      "zh-Hant": "零因子責任量化全部非零元素對；樣本乘積不足。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "finite_integral_domain",
      "role_zh": "有限整環紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "finite_commutative_ring",
      "role_zh": "父交換環",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-commutative-ring",
      "title_zh": "有限交換環",
      "reason_zh": "整環建立在同一完整交換環表。"
    },
    {
      "id": "mko-proposition",
      "title_zh": "命題",
      "reason_zh": "無零因子以蘊涵形式表述。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "責任量化全部非零元素對。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "1不等於0由父含一環保留。",
      "完整搜尋非零零因子可給資格或反例。",
      "有限整環中乘非零元是單射，有限性再推出滿射與逆元。"
    ],
    "common_misconception_zh": "抽查乘積後宣稱無零因子，或把非零元可逆未經有限性步驟直接塞進定義。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只分類指定完整有限交換環；不外推一般無限環或直接宣稱域而省略有限性定理。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "以有限性證每個非零元可逆並加入有限域。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-integral-domain.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-10-29 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}