{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-quotient-ring-01",
  "mko_id": "mko-finite-quotient-ring",
  "title": {
    "zh-Hant": "Z/4Z 對偶數理想取商",
    "en": "Quotienting Z/4Z by its even ideal"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-quotient-ring-01",
    "source": {
      "mko_id": "mko-finite-quotient-ring",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216",
      "english_sha256": "b144945c75f4a276442029de263e7ef036043832247b0b951049af0cc80623fa"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "I={0,2}為proper雙邊理想，R/I有兩個cosets及良定雙表。",
      "en": "I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "R=Z/4Z，I={0,2}；父環運算為模4加乘。",
          "en": "R=Z/4Z, I={0,2}, with addition and multiplication modulo 4."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "R=\\{0,1,2,3\\}",
        "after_tex": "\\begin{array}{c|cccc}+&0&1&2&3\\\\\\hline 0&0&1&2&3\\\\1&1&2&3&0\\\\2&2&3&0&1\\\\3&3&0&1&2\\\\\\end{array}\\qquad\\begin{array}{c|cccc}*&0&1&2&3\\\\\\hline 0&0&0&0&0\\\\1&0&1&2&3\\\\2&0&2&0&2\\\\3&0&3&2&1\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出父環全部32格。整數環法則因模4同餘與加乘相容而下降。",
          "en": "All 32 parent-ring entries are listed. Integer ring laws descend because congruence modulo 4 respects addition and multiplication."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "加法零0與乘法一1不同；不能只保留商表來取代父環。",
            "en": "Additive zero 0 and multiplicative one 1 are distinct; quotient tables do not replace the parent tables."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "I=\\{0,2\\}",
        "after_tex": "0+0=0,\\ 0+2=2,\\ 2+0=2,\\ 2+2=0;\\quad -0=0,-2=2",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "列出I全部加法與逆元，證加法子群。",
          "en": "Every sum and inverse in I is listed, proving it is an additive subgroup."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "元素2的逆元仍在I；非空與加法封閉均已核。",
            "en": "The inverse of 2 remains in I; nonemptiness and additive closure are both checked."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "r\\in R,\\quad i\\in I",
        "after_tex": "r0=0r=0;\\quad(0,1,2,3)2=(0,2,0,2)=2(0,1,2,3)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "所有左右乘積落在I。1不在I，因此I是proper雙邊理想。",
          "en": "Every left and right product lies in I. Since 1 is outside I, this is a proper two-sided ideal."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "左右兩邊皆核；不能僅由I大小就判為理想。",
            "en": "Both sides are checked; cardinality alone cannot certify an ideal."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03"
        ],
        "before_tex": "a\\sim b\\iff a-b\\in I",
        "after_tex": "E=0+I=2+I=\\{0,2\\},\\quad O=1+I=3+I=\\{1,3\\}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "同餘即同奇偶，完整列出全部cosets並去重。",
          "en": "Ideal congruence is parity equality; all cosets are listed and deduplicated."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "E與O互斥且聯集是R，商元素不是單一0或1。",
            "en": "E and O are disjoint and cover R; quotient elements are not single representatives 0 or 1."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S02",
          "S04"
        ],
        "before_tex": "a'=a+i,\\quad b'=b+j,\\quad i,j\\in I",
        "after_tex": "(a'+b')-(a+b)=i+j\\in I",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "換任意同類代表後，和只差理想元素，因此加法良定。",
          "en": "Changing either representative changes the sum by an ideal element, proving well-defined addition."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "i、j可取I全部元素，不只固定一組代表。",
            "en": "i and j range over all of I, not just a preferred representative pair."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S02",
          "S03",
          "S04"
        ],
        "before_tex": "a'=a+i,\\quad b'=b+j",
        "after_tex": "a'b'-ab=aj+ib+ij\\in I",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "左右吸收與加法封閉使三項全在I，證乘法對所有代表元良定。",
          "en": "Two-sided absorption and additive closure put all three terms in I, proving representative-independent multiplication."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "非交換父環也須保留乘積順序aj、ib；此例父環雖交換仍不能略理想條件。",
            "en": "The order aj,ib matters even in a noncommutative parent; commutativity here does not remove the ideal obligation."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S01",
          "S04",
          "S05",
          "S06"
        ],
        "before_tex": "R/I=\\{E,O\\}",
        "after_tex": "\\begin{array}{c|cc}+&E&O\\\\\\hline E&E&O\\\\O&O&E\\\\\\end{array}\\qquad\\begin{array}{c|cc}*&E&O\\\\\\hline E&E&E\\\\O&E&O\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "以完整父表計每對類的結果；前兩步證所有代表替換均保持此類。",
          "en": "Compute every pair of classes from the parent tables; the preceding proofs cover all representative substitutions."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S04",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "商表各有四格，E是零類，O是一類。",
            "en": "Each quotient table has four entries; E is the zero class and O the one class."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S03",
          "S05",
          "S06",
          "S07"
        ],
        "before_tex": "0+I=E,\\quad1+I=O",
        "after_tex": "E\\ne O,\\quad R/I\\text{ is a unital quotient ring}",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "1不在I給商中1≠0；環恆等式由良定商運算繼承。",
          "en": "Because 1 is not in I, zero and one remain distinct; ring identities descend through well-defined quotient operations."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "S05",
            "S06",
            "S07"
          ],
          "explanation": {
            "zh-Hant": "若I=R只剩一類且1=0，會違反本專案含一環慣例。",
            "en": "If I=R only one class remains and 1=0, violating this project's unital-ring convention."
          }
        },
        "leaf_basis": "mko_definition"
      }
    ],
    "result": {
      "zh-Hant": "I={0,2}為proper雙邊理想，R/I有兩個cosets及良定雙表。",
      "en": "I={0,2} is a proper two-sided ideal; R/I has two cosets and well-defined operation tables."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "R/I={0,1}作為父環子集。",
          "en": "R/I={0,1} as a subset of the parent."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "載體是E、O兩個coset集合；0、1只是可選代表。",
          "en": "Its carrier is the two coset sets E,O; 0,1 are merely chosen representatives."
        }
      },
      {
        "claim": {
          "zh-Hant": "I=R也符合本專案1≠0要求。",
          "en": "I=R also satisfies this project's 1 distinct from 0 convention."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "此時0+I=1+I=R。",
          "en": "Then 0+I=1+I=R."
        }
      },
      {
        "claim": {
          "zh-Hant": "proper理想的商必為域。",
          "en": "Every proper-ideal quotient is a field."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "Z/4Z的零理想proper，但商仍有非零2且2²=0。",
          "en": "The zero ideal of Z/4Z is proper, but its quotient retains nonzero 2 with 2^2=0."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設整數同餘與環法則；理想需加法子群及左右吸收。",
          "en": "Assume integer congruence and ring laws; an ideal is an additive subgroup absorbing multiplication from both sides."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "a約b iff a-b屬I。classes為加法cosets；[a]+[b]=[a+b]、[a][b]=[ab]。完整核代表元獨立。本專案含一環要求1不等於0，因此排除I=R的退化zero ring。",
          "en": "Define a equivalent to b when a−b belongs to I. The classes are additive cosets, with [a]+[b]=[a+b] and [a][b]=[ab]. Check representative independence completely. Because this project requires 1 distinct from 0 in a unital ring, the degenerate zero ring from I=R is excluded."
        },
        "source": {
          "mko_id": "mko-finite-quotient-ring",
          "sha256": "8c2b45b38428b422d01b869c06eaca3a0db6e14e6fafc691ef583d4614d39216",
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      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "even_product",
        "lhs": "(3*2)%4",
        "rhs": "2",
        "step": "S03"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整四元素父環與二元素商環，不把任意proper理想商都稱域。",
      "en": "A complete four-element parent ring and two-element quotient; no claim that every proper-ideal quotient is a field."
    },
    "md_symbols": [
      [
        "R",
        "模4父環",
        "parent ring modulo 4"
      ],
      [
        "I",
        "偶數理想",
        "even ideal"
      ],
      [
        "E,O",
        "完整加法cosets",
        "complete additive cosets"
      ]
    ],
    "md_error_controls": [
      {
        "title": "理想先行 / Ideal first",
        "zh": "先核加法子群與雙邊吸收，再定義商乘法。",
        "en": "Check additive subgroup and two-sided absorption before quotient multiplication."
      },
      {
        "title": "代表替換 / Representative changes",
        "zh": "a'b'−ab三項均須落I；只核0、1代表的四格不夠。",
        "en": "All three terms of a'b'-ab must lie in I; four preferred-representative products alone are insufficient."
      }
    ],
    "md_open_items": []
  },
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