❚❘×❮◆● ✣❸■ ❍➴❈ ✣➬◆● ◆❆■ ❑❍❖❆ ❙× P❍❸▼ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ◆❍➶▼ ✻ ▲❰P ❙× P❍❸▼ ❚❖⑩◆ ❆✲❑✺ ▼➷✣❯◆ ◆❐■ ❳❸ ❍å❝ ♣❤➛♥✿ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ợ P r ỗ ✲ ✷✵✶✽ ❚❘×❮◆● ✣❸■ ❍➴❈ ✣➬◆● ◆❆■ ❑❍❖❆ ❙× P❍❸▼ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❍❖⑨◆● ❚❍➚ ▼❆■ ❆◆❍ ❚➷ ❚❍➚ ì ò ◆●❯❨➍◆ ❱❹◆ ❚❘❆◆● ▼➷✣❯◆ ◆❐■ ❳❸ ❍å❝ ♣❤➛♥✿ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ▲ỵ♣✿ ✣❍❙P ❚♦→♥ ❆❑✺ ●❱❍❉✿ ◆❣✉②➵♥ r ỗ ử ♣❤➛♥ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✸ ✷ ▼æ✤✉♥ ♥ë✐ ①↕ ✺ ✶✳✶ ◆❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶ ▼æ✤✉♥ ♥ë✐ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ❚➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ♥ë✐ ①↕ ✳ ✳ ✳ ✳ ✷✳✶✳✸ ▼æ✤✉♥ ♥ë✐ ①↕ ✈➔ ♠æ✤✉♥ ❝❤✐❛ ✤÷đ❝ ✷✳✷ ❇➔✐ t➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✺ ✺ ✽ ✶✵ ✶✶ ✶✺ ✶ ▲í✐ ♠ð ✤➛✉ ❍✐➺♥ ♥❛② ❝â r➜t ♥❤✐➲✉ ♥❣÷í✐ q t ự ỵ tt ổ ố ợ ỳ s ữủ ổ ỵ t❤✉②➳t ♠ỉ✤✉♥ t↕♦ ❝ì ❤ë✐ ✤➸ t✐➳♣ ❝➟♥ s➙✉ ❤ì♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t❤➯♠ ✈➲ ♠↔♥❣ ❦✐➳♥ t❤ù❝ ♥➔②✳ ❚r♦♥❣ ✤â ♠ỉ✤✉♥ ♥ỉ✐ ①↕ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧ỵ♣ ♠ỉ✤✉♥ q trồ ữủ ú ỵ ự ỡ ởt s ổ ỵ tt ổ t❤➻ ✈✐➺❝ t➻♠ tá✐ ♥❣❤✐➯♥ ❝ù✉ ❧➔ ✤✐➲✉ r➜t ❝➛♥ t❤✐➳t ✈➔ ♥➯♥ ❧➔♠✳ ❉♦ ✤â t✉② ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ữ ợ ự t ♠ỉ✤✉♥ ♥ë✐ ①↕ ❝❤ó♥❣ tỉ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐ ✏▼➷✣❯◆ ◆❐■ ❳❸✑✳ P❤➛♥ ❜➔✐ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ♣❤➙♥ t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ■✿ ❑✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ❈❤÷ì♥❣ ■■✿ ▼ỉ✤✉♥ ♥ë✐ ①↕✳ ❈❤ó♥❣ tỉ✐ ①✐♥ tr➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü ữợ t t t ❚r➼ ✈➔ sü õ♥❣ ❤ë✱ ❝❤✐❛ s➫ t➔✐ ❧✐➺✉ tø tr ợ trữớ ỗ P ự ỵ ❞♦ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ t❤✐➳✉ sât✳ ❈❤ó♥❣ tổ ữủ sỹ õ õ ỵ t➜t ❝↔ ♠å✐ ♥❣÷í✐ ✤➸ ♣❤➛♥ ♥❣❤✐➯♥ ❝ù✉ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✷ ❈❤÷ì♥❣ ✶ ❈→❝ ♣❤➛♥ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✶✳✶ ◆❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ✭◆❤â♠ ❣✐❛♦ ❤♦→♥ ✮ ▼ët ♥❤â♠ ❣✐❛♦ ❤♦→♥ ❤❛② ♥❤â♠ ❆❜❡❧ ❧➔ ♠ët ♥❤â♠ t❤ä❛ ♠➣♥ t❤➯♠ ✤✐➲✉ ❦✐➺♥ ❧➔ ♣❤➨♣ t♦→♥ ❤❛✐ ♥❣æ✐ ❝â t❤➯♠ t➼♥❤ ❣✐❛♦ ❤♦→♥✳ ▼ët ♥❤â♠ ❣✐❛♦ ❤♦→♥ ❧➔ ♠ët t➟♣ ❤đ♣ G✱ ❝ị♥❣ ợ ởt t ổ tứ G ì G ✈➔♦ G t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ✶✳ ❚➼♥❤ ❦➳t ❤ñ♣✿ (a ∗ b) ∗ c = a ∗ (b ∗ c) ✈ỵ✐ ♠å✐ a, b ✈➔ c tở G P tỷ ỡ tỗ t ♥❤➜t ♠ët ♣❤➛♥ tû ❣å✐ ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à ỵ s ợ tỷ a t❤✉ë❝ G t❤➻ a ∗ = ∗ a = a✳ ✸✳ P❤➛♥ tû ♥❣❤à❝❤ ✤↔♦✿ ✈ỵ✐ ♠é✐ tỷ a tở G tỗ t t ởt ♣❤➛♥ tû x✱ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ a✱ s❛♦ ❝❤♦ x ∗ a = a ∗ x = 1✳ ✹✳ ❚➼♥❤ ❣✐❛♦ ❤♦→♥✿ a ∗ b = b ∗ a ✈ỵ✐ ♠å✐ a, b t❤✉ë❝ G✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ✭▼✐➲♥ ♥❣✉②➯♥ ✮ ▼✐➲♥ ♥❣✉②➯♥ ❧➔ ✈➔♥❤ ❣✐❛♦ õ ỡ ổ õ ữợ ổ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ✭❱➔♥❤ ❝❤➼♥❤ ✮ ▼✐➲♥ ♥❣✉②➯♥ X ❣å✐ ❧➔ ✈➔♥❤ ❝❤➼♥❤ ♥➳✉ ♠å✐ ✐❞❡❛❧ ❝õ❛ ♥â ✤➲✉ ❧➔ ✤÷đ❝ s✐♥❤ tø ♠ët ♣❤➛♥ tû✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❈❤♦ {Mα}α∈∆ (∆ = ∅) ❧➔ ♠ët ❤å ❝→❝ R✲ ♠æ✤✉♥✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ✤à♥❤ ♥❣❤➽❛ s❛✉✿ ✭✐✮ ▼æ✤✉♥ α∈∆ Mα ✤÷đ❝ ❣å✐ ❧➔ t➼❝❤ trü❝ t✐➳♣ ❝õ❛ ❤å ❝→❝ R✲ ♠æ✤✉♥ {Mα }α∈∆ ✳ ✭✐✐✮ ▼æ✤✉♥ α∈∆ Mα ✤÷đ❝ ❣å✐ ❧➔ tê♥❣ trü❝ t✐➳♣ ♥❣♦➔✐ ❝õ❛ ❤å ❝→❝ R✲ ♠æ✤✉♥ {Mα }α∈∆ ✳ ✭✐✐✐✮ ⑩♥❤ ①↕ β : α∈∆ Mα → Mβ ①→❝ ✤à♥❤ ❜ð✐ πβ ((xα )) = xβ ✈ỵ✐ ♠å✐ (xα ∈ α∈∆ Mα ❧➔ ♠ët t♦➔♥ ❝➜✉ ✈➔ t❛ ❣å✐ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝❤➼♥❤ t➢❝ t❤ù β✳ ✸ ❈❍×❒◆● ✶✳ ✹ ❈⑩❈ P❍❺◆ ❑■➌◆ ❚❍Ù❈ ▲■➊◆ ◗❯❆◆ ✭✐✈✮ ⑩♥❤ ①↕ iβ : Mβ → α∈∆ Mα ①→❝ ✤à♥❤ ❜ð✐ iβ (x) = (xα )✱ ✈ỵ✐ (xα ) = 0✱ ♥➳✉ α = β ✈➔ xβ = x✱ ❧➔ ✤ì♥ ❝➜✉ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ♣❤➨♣ ♥❤ó♥❣ ❝❤➼♥❤ t➢❝ t❤ù β ✳ ◆➳✉ M = ∆ Mα t❤➻ t❛ ❣å✐ M ❧➔ tê♥❣ trü❝ t✐➳♣ tr♦♥❣ ❝õ❛ ❤å {Mα }∆ ′ ◆➳✉ N ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✈➔ ❝â ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ N ❝õ❛ M s❛♦ ′ ′ ′ ❝❤♦ N N = ✈➔ M = N + N t❤➻ M = N N ✈➔ N ✤÷đ❝ ❣å✐ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ M ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ✭❉➣② ❦❤ỵ♣✮ ❉➣② Rổ ỗ R f g ổ A B C ữủ ợ t B ♥➳✉ Imf = Kerg ✳ ❉➣② ❝→❝ R✲♠æ✤✉♥ ✈➔ ỗ Rổ fn1 fn Mn1 Mn −→ Mn+1 −→ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➣② ❦❤ỵ♣ ♥➳✉ Imfn−1 = Kerfn ✈ỵ✐ ♠å✐ n ∈ Z ✳ ❉➣② ❦❤ỵ♣ −→ M −→ N −→ P ữủ ợ f g ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ✭❉➣② ❦❤ỵ♣ ❝❤➫✮ ❉➣② ❦❤ỵ♣ ♥❣➢♥ −→ A −→ B −→ ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❦❤ỵ♣ ❝❤➫ ♥➳✉ Imf ❧➔ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ B tự tỗ t ổ B ′ ❝õ❛ B s❛♦ ❝❤♦ B = Imf ⊕ B ′ ✳ ❈❤÷ì♥❣ ✷ ▼ỉ✤✉♥ ♥ë✐ ①↕ ✷✳✶ ▼ỉ✤✉♥ ♥ë✐ ①↕ ✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ▼æ✤✉♥ J ❧➔ ♠ỉ✤✉♥ ♥ë✐ ①↕ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠é✐ ✤ì♥ : A B ộ ỗ f : A J tỗ t ỗ g : B → J s❛♦ ❝❤♦ f = gα✳ ❇ð✐ α ❧➔ ✤ì♥ ❝➜✉ ♥➯♥ t❛ ❝â t❤➸ ①❡♠ ♥❤÷ A ⊂ B ✱ ✈➔ ❞♦ ✈➟② g ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ sü ♠ð rë♥❣ ❝õ❛ ❝õ❛ f tr B ỵ õ õ ữớ t❛ ①❡♠ ♠æ✤✉♥ ♥ë✐ ①↕ J ❧➔ ♠æ✤✉♥ ❝❤♦ ♣❤➨♣ sỹ rở t ỗ f : A J t ỗ g : B J ✱ tr➯♥ ♠é✐ ♠æ✤✉♥ B ⊃ A✳ G A f α G B g J ❱➼ ❞ư ✷✳✶✳✶✳ ❉➵ t❤➜② r➡♥❣✱ ♠é✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì V tr➯♥ tr÷í♥❣ sè K ✤➲✉ ❧➔ ♠ët K−♠ỉ✤✉♥ ♥ë✐ ①↕ ✈➻ ♠é✐ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ tø ♠ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❝♦♥ ❝õ❛ W ✤➳♥ V ✤➲✉ ❝â t❤➸ ♠ð rë♥❣ r❛ t♦➔♥ ❦❤ỉ♥❣ ❣✐❛♥ W ✳ ❱➼ ❞ư ✷✳✶✳✷✳ ❈❤♦ Z ❧➔ ♠ët Z−♠æ✤✉♥ ✈➔ 2Z ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ Z✳ ❑❤✐ ✤â 2Z ❦❤æ♥❣ ❧➔ ♥ë✐ ①↕ ✈➻ 2Z ❦❤æ♥❣ ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ Z✳ ●✐↔ sû 2Z ❧➔ ♥ë✐ ①↕✱ ❦❤✐ ✤â 2Z ❧➔ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ Z✱ ❤❛② Z = 2Z M ✈ỵ✐ M ❧➔ ♠ỉ✤✉♥ ❝♦♥ ❝õ❛ Z✳ ❱➻ ∈ Z ⇒ = 2k + m (2k ∈ 2Z, m ∈ M ) ⇒ m ❧➫ ⇒ 2m ❝❤➤♥✳ 2m ∈ 2Z ♠➔ Z = 2Z M ♥➯♥ 2Z 2m ∈ M (❞♦ m ∈ M ) ❉♦ ✤â 2m = ⇒ m = ⇒ = 2k( ✈æ ❧➼ ✈➻ = 2k)✳ ❱➟② 2Z ❦❤æ♥❣ ❧➔ ♥ë✐ ①↕✳ ⇒ ✺ M = {0}✳ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✻ ◆❤➟♥ ①➨t ✷✳✶✳✶✳ ✭✐✮ ▼ët A−♠æ✤✉♥ I ❧➔ ♥ë✐ ①↕ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ợ ỡ : M M →♥❤ ①↕ ❝↔♠ s✐♥❤ ′ µ∗ : HomA (M, I) → HomA M , I λ → λµ ❧➔ t♦➔♥ →♥❤✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✈➔♥❤ A ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ t❤➻ HomA (M, I) ✈➔ HomA M , I trð t❤➔♥❤ ❝→❝ A−♠ỉ✤✉♥✳ ❑❤✐ ✤â µ∗ ❧➔ ♠ët t♦➔♥ ❝➜✉ A−♠æ✤✉♥✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ♠ët A−♠æ✤✉♥ I ❧➔ ♥ë✐ ①↕ ỗ M G G M θ I ✤➲✉ ❝â t❤➸ ♥❤ó♥❣ ✈➔♦ ởt ỗ G M } µ G M λ I ✭✐✐✮ ◆➳✉ I ❧➔ ♠ët A−♠æ✤✉♥ ♥ë✐ ①↕ ✈➔ M ⊆ M ❧➔ ❝→❝ Aổ t ỗ Aổ tứ M I rở ữủ t ởt ỗ Aổ M ✤➳♥ I ✳ ❚➼♥❤ ❝❤➜t ♥➔② ✤➦❝ tr÷♥❣ ❝❤♦ ♠ỉ✤✉♥ ♥ë✐ ①↕✳ ′ ′ ❚✐➯✉ ❝❤✉➞♥ ❇❛❡r✿ R−♠æ✤✉♥ J ❧➔ ♥ë✐ ①↕ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ❜➜t ❦➻ ✐❞❡❛♥ tr I R t ý ỗ f : I J ổ ổ tỗ t tû q ∈ J s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ λ ∈ I, t❛ ❝â f (λ) = λq✳ ◆â✐ ❝→❝❤ ❦❤→❝✱ ỗ f : I J õ t rở ữủ tợ ỗ g : R → J ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû J ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕ t❤➻ ✈ỵ✐ ❜➜t ❦➻ ✐❞❡❛♥ tr→✐ I ⊂ R t ỗ f : I J tỗ t ỗ g rở f tø I ❧➯♥ t♦➔♥ R✳ ▲➜② q = g(1)✱ t❤➻ ✈ỵ✐ ♠å✐ λ ∈ I t❛ ❝â✿ f (λ) = g(λ.1) = λ.g(1) = λq ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ t❛ ❝➛♥ sû ❞ư♥❣ ❜ê ✤➲ ❩♦r♥ s❛✉ ✤➙②✿ ❇ê ✤➲ ❩♦r♥✿ ◆➳✉ tr♦♥❣ t➟♣ ❤ñ♣ s➢♣ t❤ù tü ♠➔ ♠é✐ t➟♣ ❝♦♥ s➢♣ t♦➔♥ ♣❤➛♥ ❝õ❛ ♥â ✤➲✉ ❝â ❝➟♥ tr➯♥ t❤➻ tr♦♥❣ t➟♣ ❤ñ♣ ✤â ❝â ♣❤➛♥ tû tè✐ ✤↕✐✳ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✼ ❈❤♦ A✱ B ❧➔ ❝→❝ ♠æ✤✉♥ ♠➔ A ⊂ B f : A J ỗ ❝❤ù♥❣ tä t➼♥❤ ♥ë✐ ①↕ ❝õ❛ J t❛ ❝❤➾ r❛ sỹ tỗ t rở g : B J ✳ ❳➨t ❤å D ❝→❝ ❝➦♣ ✭D✱fD ✮ tr♦♥❣ ✤â D ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ B ✱ D ⊃ A ✈➔ fD : D → J ❧➔ ♠ð rë♥❣ ❝õ❛ f : A → J ✳ ❚❛ ❝â D = ∅✳ ❚❛ s➢♣ ①➳♣ D t❤❡♦ q✉❛♥ ❤➺ t❤ù tü s❛✉✿ (D, fD ) ≥ (C, fC ) ⇔ C ⊂ D ✈➔ fD ❧➔ ♠ð rë♥❣ ❝õ❛ fC ✳ ❚❛ ❝❤➾ r❛ r➡♥❣ ❤å D ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü tr➯♥ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❜ê ✤➲ ❩♦r♥✳ ❚❤➟t ✈➟②✱ ♥➳✉ ξ ❧➔ ❜ë ♣❤➟♥ ❦❤→❝ ré♥❣ ✈➔ ✤÷đ❝ s➢♣ t♦➔♥ ♣❤➛♥ ❝õ❛ D✱ ❦❤✐ ✤â ♠ỉ✤✉♥ ❝♦♥ E = ∪D∈ξ D ✈➔ fE : E → J ♠➔ tr➯♥ ♠é✐ D ∈ ξ t❤➻ fE trò♥❣ ✈ỵ✐ fD ❧➟♣ t❤➔♥❤ ❝➦♣ ✭E, fE ✮ ❧➔ ❝➟♥ tr➯♥ ❝õ❛ ξ ✳ ❚❤❡♦ ❜ê ✤➲ ❩♦r♥✱ tr♦♥❣ D tỗ t tỷ tố G, fG t s➩ ❝❤ù♥❣ tä r➡♥❣ G = B ✳ ✈➔ ❞♦ ✤â fG ❧➔ g ❝➛♥ t➻♠✳ ❚❛ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐✱ ❦❤✐ ✤â B \ G = φ ✈➔ ❞♦ õ tỗ t x0 B \ G ổ ❝♦♥ H = G + Rx0 = {a + rx0 | a ∈ G, r ∈ R} ⊂ B ✳ ❚❛ ❝â H ⊃ G ✈➔ H = G✳ ❳➨t t➟♣ I = {λ ∈ R | λx0 ∈ G}✳ t I tr R ỗ tớ →♥❤ ①↕ h : I → J ✱ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝✿ h(λ) = fG (λx0 ), λ ∈ I, ỗ õ t t r tỗ t tỷ q J s ❝❤♦ h(λ) = λq ❇➙② ❣✐í t❛ ①➙② ❞ü♥❣ →♥❤ fH : H J ữ s ợ ộ x = a + rx0 ∈ H t❤➻ ✿ fH (x) = fG (a) + rq ủ ỵ fH ✤÷đ❝ s✉② r❛ tø ❝→❝❤ ①→❝ ✤à♥❤ ♣❤➛♥ tû q✳ ❚❤➟t ✈➟②✱ ♥➳✉ ♣❤➛♥ tû x ∈ H ❝â ✷ ❝→❝❤ ❜✐➸✉ ❞✐➵♥ ❝õ❛ x = a1 + r1x0 = a2 + r2x0 t❤➻ a1 − a2 = (r2 − r1 )x0 ∈ G✳ ❉♦ ✤â fG(a1 − a2) = fG[(r2 − r1)x0] = h(r2 − r1) = (r2 − r1)q ❱➟②✿ fG(a1)+r1q = fG(a2)+r2q✱ tù❝ fH (x) ❧➔ ❞✉② ♥❤➜t ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❝→❝❤ ❜✐➵✉ ❞✐➵♥ ❝õ❛ x ∈ H = G + Rx0✳ ❚❛ ❦✐➸♠ tr❛ ữủ fH ỗ ữ t õ (H, fH ) D ỗ tớ (H, fH ) tỹ sỹ ợ ỡ G, fG ì ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✽ ✣✐➲✉ ✤â ♠➙✉ t❤✉➝♥ ✈ỵ✐ t➼♥❤ tè✐ ✤↕✐ ❝õ❛ ❝➦♣ ✭G, fG ✮ tr♦♥❣ D ✈➔ ❞♦ ✈➟② ❜✉ë❝ ♣❤↔✐ ❝â G = B ✱ tù❝ fD = g ❝➛♥ t➻♠✳ ✷✳✶✳✷ ❚➼♥❤ ❝❤➜t ❝õ❛ ♠æ✤✉♥ ♥ë✐ ①↕ ✣à♥❤ ❧➼ ✷✳✶✳✷✳ ❚➼❝❤ trü❝ t✐➳♣ ❝õ❛ ❤å ♠æ✤✉♥ J = k∈K ❝❤➾ ❦❤✐ ♠é✐ ♠æ✤✉♥ t❤➔♥❤ Jk ự rữợ t J Jk ❧➔ ♥ë✐ ①↕ ❦❤✐ ✈➔ ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✱ t❛ ❝➛♥ ❝❤ù♥❣ tä = k∈K ♠å✐ t❤➔♥❤ ♣❤➛♥ Jt ✤➲✉ ❧➔ ♥ë✐ ①↕✱ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❇❛❡r✳ ●✐↔ sỷ f : I Jt ỗ tứ ✐✤➯❛♥ tr→✐ I ❝õ❛ R ✈➔♦ Jt✳ ◆è✐ ❦➳t f ợ ú jt : Jt Jk t ữủ ỗ jt f : I J J ổ tỗ t tỷ x ∈ J ♠➔ ✈ỵ✐ ♠å✐ λ ∈ I : jt f (λ) = λx ❑❤✐ ✤â ✈ỵ✐ ♣❤➛♥ tû xt = pt(x) ∈ Jt✱ t❛ ❝â✿ f (λ) = pt [jt f ] (λ) = pt (λx) = λpt (x) = λx ✈ỵ✐ ♠é✐ λ ∈ I ✳ ❱➟② Jt t❤ä❛ t✐➯✉ ❝❤✉➞♥ ❇❛❡r✱ tù❝ Jt ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✳ ❇➙② ❣✐í ♥➳✉ ♠å✐ ♠ỉ✤✉♥ t❤➔♥❤ ♣❤➛♥ Jk ❧➔ ♥ë✐ ①↕ ✈➔ f : I → J = Jk ❧➔ ỗ tứ tr I R J õ ợ k K ỗ ❝➜✉ fk = pk f : I → Jk ✱ Jk ổ tỗ t tû xk ∈ Jk s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ λ ∈ I : fk (λ) = λxk ✳ ❈❤å♥ ♣❤➛♥ tû x = (xk )k ∈ K ❝õ❛ J = Jk ✱ t❛ ❝â✿ f (λ) = (pk f (λ)) = (fk (λ)) = (λxk ) = λ(xk ) = λx, ∀λ ∈ I ❱➟② J t❤ä❛ ♠➣♥ t✐➯✉ ❝❤✉➞♥ ❇❛❡r✱ tù❝ J ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✳ ✣à♥❤ ❧➼ ✷✳✶✳✸✳ ❈❤♦ I t÷ì♥❣ ✤÷ì♥❣✿ ❧➔ ♠ët A−♠ỉ✤✉♥✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✭✐✮ I ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✳ f g ✭✐✐✮ ▼å✐ ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ A−♠ỉ✤✉♥ → I → M → M n → ✤➲✉ ❝❤➫ r❛✳ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✾ ✭✐✐✐✮ I ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ♠ët A−♠ỉ✤✉♥ ♥ë✐ ①↕✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ⇒ (ii) ●✐↔ sû I ❧➔ ♥ë✐ ①↕ ✈➔ ❝â ♠ët ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ A ♠æ✤✉♥ f g → I → M → M t ỗ f G I G M i I ❱ỵ✐ i ❧➔ →♥❤ ỗ t I tỗ t ởt ỗ Aổ h:M I tọ hf = i✱ ✤✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ ❞➣② ❦❤ỵ♣ ♥❣➢♥ ✤➣ ❝❤➫ r❛✳ (ii) ⇒ (iii) ●å✐ E ❧➔ ♠ët A−♠ỉ✤✉♥ ♥ë✐ ①↕ ❝❤ù❛ I ✳ ❳➨t ❞➣② ❦❤ỵ♣ ♥❣➢♥ tü ♥❤✐➯♥✿ → I → E → E/I → ❚❤❡♦ (ii) ❞➣② ❦❤ỵ♣ ♥➔② ❝❤➫ r❛✳ ❱➻ ✈➟② I ❧➔ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ E ✳ (iii) ⇒ (i) ●✐↔ sû I ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♠ët ❤↕♥❣ tû trü❝ t✐➳♣ J ❝õ❛ ♠ët A−♠æ✤✉♥ ♥ë✐ ①↕ E ✳ ❚❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ J ❧➔ ♠ët A−♠æ✤✉♥ ♥ë✐ ①↕✳ ●✐↔ sû f : V → J ❧➔ ởt ỗ g : V W ởt ỡ Aổ õ ỗ f ′ = jf : V → E ✈ỵ✐ j : J → E ❧➔ ♣❤➨♣ ♥❤ó♥❣ ❝❤➼♥❤ t➢❝✳ ❱➻ E tỗ t ỗ h : W → E t❤ä❛ ♠➣♥ h′ g = f ′ ✳ t ỗ ủ t h = ph, : W → J ✈ỵ✐ p : E → J ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❝❤➼♥❤ t➢❝✳ ❉♦ pj = idJ ✱ t❛ ❝â hg = ph′ g = pf ′ = pjf = f GV g G h f } J j Ư E h W′ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✶✵ ❙✉② r❛ J ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✳ ▼➔ I ❧➔ ✤➥♥❣ ❝➜✉ ✈ỵ✐ J ♥➯♥ I ❝ơ♥❣ ♥ë✐ ①↕✳ ✷✳✶✳✸ ▼ỉ✤✉♥ ♥ë✐ ①↕ ✈➔ ♠ỉ✤✉♥ ❝❤✐❛ ✤÷đ❝ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✹✳ ✭▼ỉ✤✉♥ ❝❤✐❛ ✤÷đ❝ ✮ ❈❤♦ R ❧➔ ♠✐➲♥ ♥❣✉②➯♥✱ ♠ỉ✤✉♥ tr➯♥ R ❣å✐ ❧➔ ♠ỉ✤✉♥ ❝❤✐❛ ✤÷đ❝ ♥➳✉ ✈ỵ✐ ♠å✐ x ∈ X ✈➔ ♠å✐ λ ∈ R \ {0} ổ ổ tỗ t tỷ y ∈ X s❛♦ ❝❤♦ λy = x X ✣à♥❤ ❧➼ ✷✳✶✳✺✳ ◆➳✉ R ❧➔ ✈➔♥❤ ❝❤➼♥❤ t❤➻ ♠å✐ R−♠æ✤✉♥ ❝❤✐❛ ✤÷đ❝ X ✤➲✉ ♥ë✐ ①↕✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ X ❧➔ ♠ỉ✤✉♥ ❝❤✐❛ ✤÷đ❝✱ I ✁ R ✈➔ f : I X ỗ r X ♥ë✐ ①↕✱ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝â ♣❤➛♥ tû q ∈ X ♠➔ ✈ỵ✐ ♠é✐ λ ∈ I t❤➻ f (λ) = λq ✳ ❇ð✐ R ❧➔ ✈➔♥❤ ❝❤➼♥❤✱ tù❝ ❧➔ ♠é✐ ✐✤➯❛♥ ❝õ❛ R ❧➔ ✐✤➯❛♥ ❝❤➼♥❤✱ ♥â✐ r✐➯♥❣ tỗ t a R I = Ra, ∈ I, ∃r ∈ R : λ = ra✳ ❑❤✐ ✤â ❝❤å♥ q ∈ X ❧➔ ♣❤➛♥ tû ♠➔ f (a) = aq X ổ ữủ ợ ♠é✐ λ ∈ I, λ = (r ∈ R)✱ t❛ ❝â✿ f (λ) = f (ra) = rf (a) = r(aq) = (ra)q = λq ❱➟② t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❇❛❡r✱ ❳ ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕✳ ✣à♥❤ ❧➼ ✷✳✶✳✻✳ ◆➳✉ R ❧➔ ♠✐➲♥ ♥❣✉②➯♥ t❤➻ ♠å✐ R−♠æ✤✉♥ ♥ë✐ ①↕ X ữủ ự r ợ ♠å✐ x ∈ X ✱ ✈ỵ✐ ♠å✐ λ ∈ R \ {0} tỗ t y X y = x✳ ❳➨t ✐✤➯❛♥ I = λR ✱ s✐♥❤ ❜ð✐ ♣❤➛♥ tû λ ✳ ❇ð✐ R ❧➔ ♠✐➲♥ ♥❣✉②➯♥ ♥➯♥ I ❧➔ ♠ỉ✤✉♥ tü ❞♦ ✈ỵ✐ ❝ì sð ❝❤➼♥❤ ❧➔ t➟♣ ♠ët ♣❤➛♥ tû {λ}✳ ⑩♥❤ ①↕ ϕ : {λ} → X ♠➔ ϕ(λ) = x ❝â t❤➸ ♠ð rë♥❣ tỵ✐ ỗ : I X X t t r tỗ t tỷ y ∈ X s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ r ∈ I t❤➻ ϕ(r) = ry✳ ◆â✐ r✐➯♥❣✱ ❦❤✐ r = λ t❤➻✿ x = ϕ(λ) = λy ❱➟② X ❧➔ ♠æ✤✉♥ ữủ ì t➟♣ ❈▼❘ ♠ët A✲♠æ✤✉♥ I ❧➔ ♥ë✐ ①↕ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ A✲♠ỉ✤✉♥ ❇➔✐ ✷✳✷✳✶✳ f g −→ M ′ −→ M −→ M ′′ −→ ❞➣② g∗ f∗ −→ Hom(M ′′ , I) −→ Hom(M, I) −→ Hom(M ′ , I) −→ ✭❛✮ ✭f ∗ = Hom(f, idI ) ✈➔ g∗ = Hom(g, idI )✮ ❝ơ♥❣ ❧➔ ❦❤ỵ♣ ♥❣➢♥✳ g ′ f ′′ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ❞➣② ❦❤ỵ♣✿ −→ M −→ M −→ M −→ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❞➣② g∗ f∗ −→ Hom(M ′′ , I) −→ Hom(M, I) −→ Hom(M ′ , I) −→ ◆❤➟♥ ①➨t r➡♥❣ g∗ ❧➔ ♠ët ✤ì♥ ❝➜✉✱ ✈➻ ♥➳✉ α ∈ Kerg∗✱ tù❝ ❧➔ αg = 0✱ t❤➻ ❞♦ g ❧➔ t♦➔♥ ❝➜✉✱ t❛ s✉② r❛ α = 0✳ ❱➟② t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ Img∗ = Kerf ∗✳ f ∗ g ∗ = Hom(f, idI ).Hom(g, idI ) = Hom(gf, idI ) = Hom(0, idI ) =0 ❉♦ ✤â Img ∗ ⊆ Kerf ∗ ❚❛ ❝❤ù♥❣ ♠✐♥❤ Kerf ∗ ⊆ Img∗ ▲➜② β ∈ Hom(M, I) s❛♦ ❝❤♦ f ∗(β) = tù❝ ❧➔ βf = 0✳ ❈❤ó♥❣ t ỹ ởt ỗ : M → I t❤ä❛ ♠➣♥ g∗(γ) = β ✳ ❱ỵ✐ ♠é✐ m′′ ∈ M ′′✱ ❞♦ g ❧➔ t♦➔♥ ❝➜✉ ♥➯♥ ∃m ∈ M s❛♦ ❝❤♦ p = g(m) ❳➨t t÷ì♥❣ ù♥❣ γ : M” → I m” → γ(m”) = β(m) ✣➛✉ t✐➯♥✱ t÷ì♥❣ ù♥❣ tr➯♥ ❧➔ ♠ët →♥❤ ①↕✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû m1, m2 ∈ M s❛♦ ❝❤♦ g(m1) = g(m2) ✭✶✮ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✶✷ ❑❤✐ ✤â m1 − m2 ∈ Kerg = Imf ⊆ Kerβ ✳ ✭ ◆❤ỵ r➡♥❣ βf = 0✮ ❙✉② r❛ β(m1 ) = β(m2 )✳ ❉♦ ✤â γ ❧➔ ♠ët →♥❤ ①↕✳ ❚❛ ❦✐➸♠ tr❛ ✤÷đ❝ r➡♥❣ γ ❧➔ ♠ët ỗ tọ = g = g (γ)✳ ❱➟② β ∈ Img ∗ ❚ø ✤â s✉② r❛ Kerf ∗ ⊆ Img ∗ ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮ s✉② r❛ Img ∗ = Kerf ∗ ✳ ❱➟② ❞➣② ✭❛✮ ❧➔ ❞➣② ❦❤ỵ♣✳ ❚❛ ❝â ❞➣② ✭❛✮ ❧➔ ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ∗ ❧➔ t♦➔♥ ❝➜✉✳ ▼➦t ❦❤→❝ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ I ❧➔ ♠æ✤✉♥ ♥ë✐ ①↕ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠é✐ ✤ì♥ ❝➜✉ f ✱ →♥❤ ①↕ ❝↔♠ s✐♥❤ f ∗ ❝ô♥❣ ❧➔ t♦➔♥ ❝➜✉ ✭ ◆❤➟♥ ①➨t ✷✳✶✳✶✮ ❱➻ ✈➟② I ❧➔ ♥ë✐ ①↕ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❞➣② ✭❛✮ ❧➔ ❞➣② ❦❤ỵ♣ ♥❣➢♥✳ ▼ët ♥❤â♠ ❆❜❡❧ D ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐❛ ✤÷đ❝ ♥➳✉ ✈ỵ✐ ♠å✐ d ∈ D ✈➔ ♠å✐ sè ♥❣✉②➯♥ n = tỗ t c D s d = nc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ❇➔✐ ✷✳✷✳✷✳ ✭✐✮ ❍↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝ ❧➔ ❝❤✐❛ ✤÷đ❝✳ ✭✐✐✮ ▼å✐ tr÷í♥❣ ❝â ✤➦❝ sè ✵ ❧➔ ❝❤✐❛ ✤÷đ❝✳ ✭✐✐✐✮ ▼ët ♥❤â♠ ❛❜❡♥ ❝❤✐❛ ✤÷đ❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ ♠ët Z ✲♠æ✤✉♥ ♥ë✐ ①↕✳ ❚ø ✤â s✉② r❛ Q/Z ❧➔ ♠ët Z ✲♠æ✤✉♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❤↕♥❣ tû trü❝ t✐➳♣ ❝õ❛ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝ ❝ơ♥❣ ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ●✐↔ sû D = D1 ⊕ D2 ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ D1 ❝❤✐❛ ✤÷đ❝✳ ❱ỵ✐ ♠å✐ ♣❤➛♥ tû d1 ∈ D1 ✈➔ ♠å✐ sè ♥❣✉②➯♥ n = 0✱ ❞♦ d1 t❤✉ë❝ D ✈➔ D ữủ tỗ t c D d1 = nc✳ ❱➻ D = D1 ⊕ D2 ♥➯♥ c = c1 + c2 ✈ỵ✐ c1 ∈ D1 ✈➔ c2 ∈ D2 ❉♦ ✤â d1 = nc1 + nc2 ✈➔ ✤➙② ❧➔ ❜✐➸✉ ❞✐➵♥ ❞✉② ♥❤➜t ❝õ❛ d1 tr♦♥❣ tê♥❣ trü❝ t✐➳♣ D1 ⊕ D2 ✳ ❚ø ✤â s✉② r❛ d1 = nc1 + nc2 ✈➔ = nc2 ✳ ❱➟② ♠å✐ ♣❤➛♥ tû d1 ∈ D1 ✈➔ ♠å✐ số n = tỗ t c1 D1 ✤➸ d1 = nc1 ♥❣❤➽❛ ❧➔ D1 ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ✭✐✐✮ ●✐↔ sû X ❧➔ ♠ët trữớ õ số ợ ỡ e ❑❤✐ ✤â ne = 0✱ ❞♦ ✤â ne ❦❤↔ ♥❣❤à❝❤ ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ n = ✳ ❱ỵ✐ ♠å✐ x ∈ X ✱ ✤➦t y = (ne)−1 x t❛ ❝â ny = n(ey) = (ne)y = (ne)(ne)−1 x = x✳ ❈❍×❒◆● ✷✳ ▼➷✣❯◆ ◆❐■ ❳❸ ✶✸ ❱➟② X ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ✭✐✐✐✮ ✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ●✐↔ sû D ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ D ❧➔ ♠ët Z−♠æ✤✉♥ ♥ë✐ ①↕ ❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ t✐➯✉ ❝❤✉➞♥ ❇❛❡r✳ ●å✐ I ❧➔ ♠ët ✐❞❡❛❧ ❜➜t ❦➻ ❝õ❛ Z ♥➯♥ I = nZ ✈ỵ✐ ♠ët sè n õ sỷ õ ỗ f : Z ỹ ỗ g : Z → ❉ ❧➔ ♠ð rë♥❣ ❝õ❛ f n = t f ỗ ổ õ t g ỗ ổ t❤➻ ❤✐➸♥ ♥❤✐➯♥ g ❧➔ ♠ð rë♥❣ ❝õ❛ f ✳ ●✐↔ sû n = 0✳ ❱➻ f (n) ∈ D D ữủ tỗ t x D ✤➸ f (n) = nx✳ ❇➡♥❣ ❝→❝❤ ✤➦t g(a) = ax ợ a Z t ữủ ỗ ❝➜✉ g:Z→❉ ❉♦ g(na) = (na)x = a(nx) = af (n) = f (na) ✈ỵ✐ ♠å✐ a ∈ Z ♥➯♥ g ❧➔ ♠ð rë♥❣ ❝õ❛ f ✱ ♥❣❤➽❛ ❧➔ ❜✐➸✉ ỗ s G nZ f ~ D i G Z g ✣✐➲✉ ❦✐➺♥ ✤õ✳ ●✐↔ sû ❉ ❧➔ ♠ët Z−♠æ✤✉♥ ♥ë✐ ①↕ ✈➔ ❞ ❧➔ ♠ët ♣❤➛♥ tû ❜➜t ❦➻ ❝õ❛ ❉✳ ❱ỵ✐ sè ♥❣✉②➯♥ ♥ = ✵ ❜➜t ❦➻ t❛ ①➨t ✤ì♥ ❝➜✉ ❝❤➼♥❤ t➢❝ i : Z Z ỗ f : ♥Z → ❉ ①→❝ ✤à♥❤ ❜ð✐ f (n) = d ✭♥❣❤➽❛ ❧➔ ❢✭♥♠✮ ❂ ♠❞✱ ♠ ∈ Z✮✳ ❉♦ ❉ tỗ t ỗ g : Z → ❉ ❧➔ ♠ð rë♥❣ ❝õ❛ f ✳ ❚❛ ❝â d = f (n) = g(n) = ng(1)✳ ✣➦t c = g(1) ∈ D t❛ ♥❤➟♥ ✤÷đ❝ d = nc✳ ❱➟② D ❧➔ ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ❚r÷í♥❣ ❝→❝ sè ❤ú✉ t➾ Q ❝â ✤➦❝ sè ♥➯♥✱ t❤❡♦ ✭✐✐✳✮✱ Q ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✳ ❱➻ ✈➟② t❤❡♦ ✭✐✳✮✱ Q/Z ❧➔ ♠ët ♥❤â♠ ❆❜❡❧ ❝❤✐❛ ✤÷đ❝✱ ✈➔ ❞♦ ✤â ♥â ❧➔ ♠ët Z−♠æ✤✉♥ ♥ë✐ ①↕✳ ❑➳t ❧✉➟♥ ❚r♦♥❣ ♣❤➛♥ ♥❣❤✐➯♥ ❝ù✉ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ð ❈❤÷ì♥❣ ■ ♥❤➡♠ ♣❤ư❝ ✈ư ❜ê trđ ❝❤♦ ♣❤➛♥ ❦✐➳♥ t❤ù❝ ð ❝❤÷ì♥❣ ■■ ✈➲ ♠ỉ✤✉♥ ♥ë✐ ①↕✳ ▼ët sè ❜➔✐ t➟♣ ✤÷đ❝ ✤÷❛ r❛ ♥❤➡♠ ❝õ♥❣ ❝è ♣❤➛♥ t❤ù❝ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉✳ ✶✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ t ổ r ố ỗ ố P ỗ ❬✷❪ ❍✳❆✳◆✐❡❧s❡♥✱ ❊❧❡♠❡♥t❛r② ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ▲❡❝t✉r❡ ♥♦t❡s✱ ✷✵✵✺✳ ❬✸❪ ❏♦s❡♣❤ ❏✳❘♦t♠❛♥✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❍♦♠♦❧♦❣✐❝❛❧ ❆❧❣❡❜r❛✱ ❙♣r✐♥❣❡r✱✶✾✼✾✳ ❬✹❪ ◆❣✉②➵♥ ❚✐➳♥ ◗✉❛♥❣ ✲ ◆❣✉②➵♥ ❉✉② ❚❤✉➟♥✱ ❈ì ❙ð ▲➼ ❚❤✉②➳t ▼ỉ✤✉♥ ✈➔ ✈➔♥❤✱ ◆❳❇ ●✐→♦ ❉ư❝✱ ✷✵✵✶✳ ❬✺❪ ❚r÷ì♥❣ ❈ỉ♥❣ ◗✉ý♥❤ ✲ ▲➯ ❱➠♥ ❚✉②➳t✱ ●✐→♦ tr➻♥❤ ❧➼ t❤✉②➳t ✈➔♥❤ ✈➔ ♠ỉ✤✉♥✱ ◆❳❇ ✣↕✐ ❍å❝ ❍✉➳✱ ✷✵✶✸✳ ❬✻❪ ❉÷ì♥❣ ◗✉è❝ ❱✐➺t ✲ ▲➯ ❱➠♥ ✣➼♥❤ ✲ ✣➦♥❣ ✣➻♥❤ ❍❛♥❤ ✲ ✣➔♦ ổ rữỡ ỗ ❚❤❛♥❤ ✲ P❤❛♥ ❚❤à ❚❤õ②✱ ❇➔✐ ❚➟♣ ▲➼ ❚❤✉②➳t ▼♦❞✉❧❡✱ ◆❳❇ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✱ ❬✼❪ ❉÷ì♥❣ ◗✉è❝ ❱✐➺t✱❈ì sð ▲➼ ❚❤✉②➳t ▼♦❞✉❧❡✱ ◆❳❇ ✣↕✐ ❍å❝ ❙÷ P❤↕♠✱ ✶✺