{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-two-sided-ideal",
  "semantic_family_id": "sf-finite-two-sided-ideal",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限雙邊理想"
  },
  "summary": {
    "zh-Hant": "有限環子集若為加法子群且吸收父環的左右乘法，則為雙邊理想。"
  },
  "statement": {
    "zh-Hant": "給有限含一環R與完整子集I。I須非空、對加法及加法逆元封閉，並對所有r屬R、x屬I同時有r乘x與x乘r屬I。資格永遠綁定同一父環。"
  },
  "formula": {
    "tex": "I(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>I</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "I",
        "glyph": "I"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ca3809d35d60a919b12b7e0e6a0de5b84bf0cf153de5f6c9efe9134881cfbc17"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "父環以完整雙表通過有限含一環資格，I由互異載體元素完整列出。"
    },
    {
      "id": "a2",
      "zh-Hant": "左右吸收分開量化；有限抽樣不證雙邊性。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "two_sided_ideal",
      "role_zh": "雙邊理想",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "finite_unital_ring",
      "role_zh": "父環",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-unital-ring",
      "title_zh": "有限含一環",
      "reason_zh": "理想運算繼承自同一父環。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "I是父載體的明示子集。"
    },
    {
      "id": "mko-finite-abelian-group",
      "title_zh": "有限阿貝爾群",
      "reason_zh": "I在加法下須成子群。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "加法子群資格包含0與加法逆元。",
      "左吸收與右吸收各自核全部父環元素。",
      "零理想與整環自身都是雙邊理想；子環或單側理想不能混稱。"
    ],
    "common_misconception_zh": "只核乘法封閉、只核左吸收，或未綁父環就稱雙邊理想。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核指定完整有限父環與子集；不形成商環，也不宣稱極大／質理想。",
    "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": "加入有限環的商環與代表元良定性。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-two-sided-ideal.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-10-28 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"
  }
}