{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-commutative-ring",
  "semantic_family_id": "sf-finite-commutative-ring",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限交換環"
  },
  "summary": {
    "zh-Hant": "有限含一環若乘法對全部元素對交換，則為有限交換環；零因子仍可能存在。"
  },
  "statement": {
    "zh-Hant": "在已通過完整有限含一環資格的同一加法與乘法表上，若每個a,b皆有a乘b=b乘a，則為有限交換環。本輪逐一核完整元素對，不由幾個樣本或表面對稱猜測。"
  },
  "formula": {
    "tex": "C(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "edb25cc2211e62a78b4da558ef318ebd833eb6f3b66a4334a6411cf8263d225b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "共同載體及完整加法／乘法表均明示，且先通過有限含一環全部公理。"
    },
    {
      "id": "a2",
      "zh-Hant": "交換律量化所有有序元素對；不額外假設無零因子或非零元可逆。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "commutative_ring_record",
      "role_zh": "交換環紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "finite_unital_ring",
      "role_zh": "父含一環",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-unital-ring",
      "title_zh": "有限含一環",
      "reason_zh": "交換性只在同一已認證環表上附加。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "乘法交換律量化全部元素對。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先保留父環的0、1與全部環公理。",
      "乘法交換律逐元素對核對。",
      "交換環仍可能有非零零因子，整環與域另立節點。"
    ],
    "common_misconception_zh": "看到若干乘積可交換，就宣稱整個環交換，或把交換環直接當整環／域。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只分類指定完整有限環；不推出無零因子、域或一般無限環結論。",
    "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": "加入有限整環與有限交換係數環上的多項式環。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-commutative-ring.md 整理；2026-10-14 脈衝 R01（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"
  }
}