{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-polynomial-ring-over-finite-commutative-ring",
  "semantic_family_id": "sf-polynomial-ring-over-finite-commutative-ring",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限交換環上的多項式環"
  },
  "summary": {
    "zh-Hant": "係數取自有限交換環的普通多項式以有限支撐係數序列表示，但所有任意有限次多項式合成的載體仍是無限。"
  },
  "statement": {
    "zh-Hant": "R為有限交換含一環時，R[x]的每個元素是有限支撐係數序列；加法逐係數，乘法為有限卷積。若1不等於0，單項式1,x,x平方等依degree互異，因此普通R[x]不稱有限環。"
  },
  "formula": {
    "tex": "P(R)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "R",
          "glyph": "R"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "418b8ad60443b41294110a73c03519dc4814d745688d483b8d2bbd86070c5885"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "係數環用完整有限交換環雙表；多項式係數陣列以尾端零正規化。"
    },
    {
      "id": "a2",
      "zh-Hant": "不假設R為域、整環、UFD或Euclidean domain；有零因子時不使用degree乘法加法公式。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "polynomial_ring_record",
      "role_zh": "多項式環紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "R",
      "role": "finite_commutative_ring",
      "role_zh": "有限係數環",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-commutative-ring",
      "title_zh": "有限交換環",
      "reason_zh": "係數加乘由父環完整表給出。"
    },
    {
      "id": "mko-polynomial",
      "title_zh": "多項式",
      "reason_zh": "元素以有限係數序列表示。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "有限支撐係數序列及卷積使用索引。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "尾端零不改多項式，須正規化。",
      "加法逐項、乘法有限卷積並用R內運算。",
      "有限係數集不限制degree，因此R[x]通常無限。"
    ],
    "common_misconception_zh": "把有限係數環上的普通多項式環也叫有限載體，或在有零因子時硬套degree(fg)=degree f+degree g。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只計算明示有限支撐多項式；不宣稱域、整環、UFD、Euclidean或有限載體。",
    "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": "加入多項式理想、商環與有限域多項式。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-polynomial-ring-over-finite-commutative-ring.md 整理；2026-10-14 脈衝 R02（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"
  }
}