{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-finite-commutative-ring"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-commutative-ring-01","title":{"zh-Hant":"有限交換環：逐元素對核對 ℤ₆ 的交換律，並以上三角矩陣環為非交換反例","en":"Finite commutative ring: checking commutativity of ℤ₆ over all element pairs, with an upper-triangular matrix ring as a non-commutative counterexample"},"summary":{"zh-Hant":"（一）在已通過有限含一環資格的 ℤ₆={0,1,2,3,4,5}（a⊕b=(a+b) mod 6、a⊗b=(ab) mod 6）上，核對全部 36 個有序元素對的 a⊗b=b⊗a，判定 ℤ₆ 是否為有限交換環。（二）說明 ℤ₆ 有非零零因子且 2 不可逆，所以不是整環或域。（三）在已通過資格的 ℱ₂ 上 2×2 上三角矩陣環 U 中，說明若干乘積可交換，但 U 不是交換環。","en":"(1) On ℤ₆={0,1,2,3,4,5} (a⊕b=(a+b) mod 6, a⊗b=(ab) mod 6), already qualified as a finite unital ring, check a⊗b=b⊗a for all 36 ordered element pairs and decide whether ℤ₆ is a finite commutative ring. (2) Show that ℤ₆ has nonzero zero divisors and that 2 is not invertible, so it is neither an integral domain nor a field. (3) In the qualified ring U of 2-by-2 upper-triangular matrices over 𝔽₂, show that several products commute and yet U is not commutative."},"mko_ids":["mko-finite-commutative-ring"]}],"title":"概念推導導讀"}