{"schema_version":"ocme-site-results-v1","locale":"en","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-unital-ring"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-unital-ring-01","title":{"zh-Hant":"有限含一環：證明模 4 的加法表與乘法表構成含一環","en":"Finite unital ring: proving that the mod-4 addition and multiplication tables form a unital ring"},"summary":{"zh-Hant":"在載體 X={0,1,2,3} 上給定完整的加法表 ⊕ 與乘法表 ⊙（即模 4 的加法與乘法）。對所有元素對與三元組證明：(X,⊕) 是阿貝爾群；⊙ 結合；⊙ 對 ⊕ 左右分配；存在唯一的雙側乘法單位元 1，且 1≠0。並指出 2⊙2=0 不妨礙它是含一環。","en":"On the carrier X={0,1,2,3}, complete addition and multiplication tables ⊕ and ⊙ (addition and multiplication mod 4) are given. Prove for all pairs and triples: (X,⊕) is an abelian group; ⊙ is associative; ⊙ distributes over ⊕ on both sides; there is a unique two-sided multiplicative identity 1, and 1≠0. Note also that 2⊙2=0 does not stop it from being a unital ring."},"mko_ids":["mko-finite-unital-ring"]}],"title":"Concept derivation guides"}