Môđun đối đồng điều địa phương cấp cao nhất và tính catenary của giá không trộn lẫn của môđun hữu hạn sinh

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖

❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆

P❍❆◆ ❚❍➚ ✣➱ ◗❯❨➊◆

▼➷✣❯◆ ✣➮■ ✣➬◆● ✣■➋❯ ✣➚❆ P❍×❒◆●
❈❻P ❈❆❖ ◆❍❻❚ ❱⑨ ❚➑◆❍ ❈❆❚❊◆❆❘❨
❈Õ❆ ●■⑩ ❑❍➷◆● ❚❘❐◆ ▲❼◆ ❈Õ❆ ▼➷✣❯◆
❍Ú❯ ❍❸◆ ❙■◆❍

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❇➻♥❤ ✣à♥❤ ✲ ◆➠♠ ✷✵✶✾


✶

▼Ð ✣❺❯
❱➔♦ ♥❤ú♥❣ ♥➠♠ ✶✾✻✵✱ ❆✳ ●r♦t❤❡♥❞✐❡❝❦ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ỵ tt ố ỗ
ữỡ ỵ tt ♥➔② ✤➣ trð t❤➔♥❤ ❝ỉ♥❣ ❝ư ❦❤ỉ♥❣ t❤➸
t❤✐➳✉ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❚♦→♥ ❤å❝ ♥❤÷ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱
❍➻♥❤ ❤å❝ ✣↕✐ sè✱ ✣↕✐ sè tê ❤ñ♣✳✳✳
▼ët tr♦♥❣ ỳ t q q trồ ổ ố ỗ ✤à❛
♣❤÷ì♥❣ ❧➔ t➼♥❤ tr✐➺t t✐➯✉ ✈➔ t➼♥❤ ❆rt✐♥✳ ❈❤♦ (R, m) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥
◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻ Hmd (M ) = 0 ✈➔
❦❤æ♥❣ ❤ú✉ ❤↕♥ s✐♥❤✱ tr♦♥❣ ✤â d = ❞✐♠M ✈➔ Hmi (M ) ❧➔ ♠ỉ✤✉♥ ❆rt✐♥ ✈ỵ✐
♠å✐ i ≥ 0.
❚❤❡♦ ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❬✶✷❪✱ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ R✲♠æ✤✉♥
❆rt✐♥ A✱ ❦➼ ❤✐➺✉ ❧➔ ❆ttR(A) ❝â ✈❛✐ trá ♥❤÷ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥
❦➳t ✤è✐ ✈ỵ✐ ♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ◆❤✐➲✉ ♥❤➔ ❚♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥

❝ù✉ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ổ ố ỗ ữỡ
t ợ ỹ Hmi (M ) ổ ố ỗ ữỡ
t HId(M ) ợ ởt ✐✤➯❛♥ I ❝õ❛ R✱ ✤➸ tø ✤â ❧➔♠ rã ❝➜✉ trú
ổ M ỡ s R rữợ t ✷✵✵✼ ◆❣✉②➵♥ ❚ü ❈÷í♥❣✱ ▲➯
❚❤❛♥❤ ◆❤➔♥ ✈➔ ◆❣✉②➵♥ ❚❤à ❉✉♥❣ ❬✼❪ ✤➣ ✤➲ ①✉➜t ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t✿


✷
❱ỵ✐ ♠é✐ R✲♠ỉ✤✉♥ ❆rt✐♥ A✱
❆♥♥R (0 :❆ p) = p ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝❤ù❛ ❆♥♥R(❆).
❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤➸ t➻♠ ❤✐➸✉ s➙✉ ❤ì♥ ✈➲ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱ ú tổ ồ
tổ ố ỗ ữỡ ❝➜♣ ❝❛♦ ♥❤➜t ✈➔ t➼♥❤

❝❛t❡♥❛r② ❝õ❛ ❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥ ❝õ❛ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✧✳

◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ t t ỗ
õ ❝❤÷ì♥❣✳

❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➢♥ t➢t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ✤➛② ✤õ❀ ✤à❛
♣❤÷ì♥❣ ❤â❛❀ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì❀ ❝❤✐➲✉ ❑r✉❧❧❀ ♠ỉ✤✉♥ ❆rt✐♥❀
tự tr ổ ố ỗ ữỡ t tr
ừ ỗ

ữỡ ổ ố ỗ ữỡ t

ở ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤ó♥❣ tỉ✐ s➩ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉
✈➲ t➼♥❤ ❝❤➜t✿ ❆♥♥(0 :❆ p) = p ✈ỵ✐ ♠å✐ p ∈ V (❆♥♥R(A))❀ t➼♥❤ ❝❤➜t ✭✯✮ ✤è✐
✈ỵ✐ ♠ỉ✤✉♥ ố ỗ ữỡ t t ❝❛t❡♥❛r② ❝õ❛

❣✐→ ❦❤æ♥❣ trë♥ ❧➝♥✳
❚➜t ❝↔ ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ tr➼❝❤ ❞➝♥ ❝❤õ ②➳✉ tø t➔✐ ❧✐➺✉
❬✼❪✳ ◆❤✐➺♠ ✈ư ❝õ❛ ❝❤ó♥❣ tỉ✐ ❧➔ ❧➔♠ rã ❧↕✐ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ✤â ✈➔ ❤➺ t❤è♥❣
❧↕✐ t❤❡♦ ♠ët ❜è ử ủ ỵ
ữủ t ữợ sỹ ữợ ừ
ỏ tớ t t tr ữợ
õ ỵ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥
♥➔②✳ ❚æ✐ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛②✳


✸
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❍å❝ ◗✉② ◆❤ì♥
❝ị♥❣ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ✈➔ ♥❣♦➔✐ ❑❤♦❛ ❚♦→♥ ✈➔ ❚❤è♥❣ ❦➯ ✤➣ t➟♥ t➻♥❤
❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣✳
❈✉è✐ ❝ị♥❣ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ ❤å❝ ✈✐➯♥ tr♦♥❣ ợ ồ
õ ỳ ữớ t❤➙♥ tr♦♥❣ ❣✐❛ ✤➻♥❤ ✤➣ ✤ë♥❣ ✈✐➯♥✱
❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥
t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
▼➦❝ ❞ò ❜↔♥ t❤➙♥ ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣ ✈➔ ♥ê ❧ü❝ ❧➔♠ ✈✐➺❝ r➜t ♥❤✐➲✉ ✤➸
❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔② ♥❤÷♥❣ ❞♦ ✤✐➲✉ ❦✐➺♥✱ t❤í✐ ❣✐❛♥✱ tr➻♥❤ ✤ë ❦✐➳♥ t❤ù❝
✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐
♥❤ú♥❣ t❤✐➳✉ sât✳ tổ rt ữủ sỹ õ ỵ ừ t❤➛②✱ ❝ỉ ❣✐→♦ ✈➔
❜↕♥ ✤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳
◆❣➔② ✷✾ t❤→♥❣ ✽ ♥➠♠ ✷✵✶✾

❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐
P❤❛♥ ❚❤à ✣é ◗✉②➯♥


✹

❈❤÷ì♥❣ ✶
❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✤➣ ❜✐➳t ✈➲ ✤➛②
✤õ❀ ✤à❛ ♣❤÷ì♥❣ ❤â❛❀ sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì❀ ❝❤✐➲✉ ❑r✉❧❧❀ ♠ỉ✤✉♥ rt
tự tr ổ ố ỗ ữỡ t
tr ừ ỗ t❤✉➟♥ t✐➺♥ ❝❤♦ ✈✐➺❝ t❤❡♦ ❞ã✐
❦➳t q✉↔ tr♦♥❣ ❝❤÷ì♥❣ s❛✉✳

✶✳✶ ✣➛② ✤õ
◆ë✐ ❞✉♥❣ ❝õ❛ t✐➳t ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② t❤❡♦ t➔✐ ❧✐➺✉ ❬✶✹❪✳ ❈❤♦ R ❧➔ ♠ët
✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼ët ✈➔♥❤ ❧å❝

❧➔ ♠ët ✈➔♥❤ R ❝ị♥❣ ✈ỵ✐ ♠ët ❤å
(Rn )n 0 ❝→❝ ♥❤â♠ ❝♦♥ ❝õ❛ R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿
(i) R0 = R❀
(ii) Rn+1 ⊂ Rn ✈ỵ✐ ♠å✐ n 0❀
(iii) Rn Rm ⊂ Rn+m ✈ỵ✐ ♠å✐ n, m 0✳
R


✺

❱➼ ❞ö ✶✳✶✳✷✳ (i) ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤✳ ▲➜② R0 = R ✈➔ Rn = 0 ✈ỵ✐ ♠å✐

❑❤✐ ✤â (Rn)n 0 ❧➔ ♠ët ❧å❝ ❝õ❛ R ✈➔ ❣å✐ ❧➔ ♠ët ❧å❝ t➛♠ t❤÷í♥❣✳
(ii) ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â (I n )n 0 ❧➔ ♠ët ❧å❝ ❝õ❛ R✱ ♥â
✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧å❝ I ✲❛❞✐❝✳

(iii) ❈❤♦ (Rn )n 0 ❧➔ ♠ët ❧å❝ ❝õ❛ R ✈➔ S ❧➔ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ R✳ ❑❤✐
✤â (Rn ∩ S)n 0 ❧➔ ♠ët ❧å❝ ❝õ❛ S ✱ ♥â ✤÷đ❝ ❣å✐ ❧➔ ❧å❝ ❝↔♠ s✐♥❤ tr➯♥ S ✳

n

1✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❧å❝ ✈ỵ✐ ❧å❝ (Rn)n 0✳ ▼ët R✲♠ỉ✤✉♥
▼ ❧å❝ ❧➔ ♠ët R✲♠ỉ✤✉♥ M ❝ị♥❣ ✈ỵ✐ ♠ët ❤å (Mn)n 0 ❝→❝ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛
M t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥✿
(i) M0 = M ❀
(ii) Mn+1 ⊂ Mn ✈ỵ✐ ♠å✐ n 0❀
(iii) Rn Mm ⊂ Mn+m ✈ỵ✐ ♠å✐ n, m 0✳

❱➼ ❞ư ✶✳✶✳✹✳

(i) ❈❤♦ M

❧➔ ♠ët R✲♠ỉ✤✉♥ ✈➔ R ❝â ❧å❝ t➛♠ t❤÷í♥❣✳ ❑❤✐ ✤â
M ❝ơ♥❣ ❝â ♠ët ❧å❝ t➛♠ t❤÷í♥❣ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ M0 = M ✈➔ Mn = 0
✈ỵ✐ ♠å✐ n 1✳
(ii) ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ ①➨t ❧å❝ I ✲❛❞✐❝ ❝õ❛ R✳ ✣à♥❤ ♥❣❤➽❛ ❧å❝
I ✲❛❞✐❝ ❝õ❛ M ❜➡♥❣ ❝→❝❤ ❧➜② Mn = I n M ✳ ❑❤✐ ✤â M ❧➔ ♠ët R✲♠æ✤✉♥ ❧å❝✳
❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥ ❧å❝✳ ▲å❝ (Mn)n 0 tr M ởt tổổ
tr M tữỡ t ợ ❝➜✉ tró❝ ♥❤â♠ ❝♦♥ ❛❜❡❧ ❝õ❛ M ♠➔ (Mn)n 0 ❧➔
♠ët ❝ì sð ❧➙♥ ❝➟♥ ❝➟♥ ❝õ❛ ✵✳ ❚ỉ♣ỉ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ ❝↔♠ s✐♥❤ ❜ð✐
❧å❝ (Mn)n 0✳ ❈❤♦ M ❧➔ ♠ët R✲♠ỉ✤✉♥ ✈ỵ✐ ❧å❝ (Mn)n 0 ✈➔ tỉ♣ỉ ữủ
ồ (Mn)n0. rữợ t ú tổ ❧↕✐ ❦❤→✐ ♥✐➺♠ ❞➣② ❈❛✉❝❤②✿
▼ët ❞➣② (xn) ❝→❝ ♣❤➛♥ tû tr M ữủ ồ ởt
ợ ộ k N tỗ t n0 s xm xn ∈ Mk , ✈ỵ✐ ♠å✐ m, n n0✳

✻
●å✐ T ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❞➣② ❈❛✉❝❤② tr♦♥❣ M ✳ ❚r➯♥ T q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐
✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐✿
❱ỵ✐ ♠å✐ (xn)✱ (yn) ∈ T, (xn) ∼ (yn) ⇔ ợ ộ m N, tỗ t n0 s
xn − yn ∈ Mm , ✈ỵ✐ ♠å✐ n ≥ n0 .
❑❤✐ ✤â q✉❛♥ ❤➺ tr➯♥ ❧➔ q✉❛♥ ❤➺ t÷ì♥❣ ✤÷ì♥❣✳ ❑➼ ❤✐➺✉
M = T /∼ = {(xn )|(xn ) ∈ T }.

❚÷ì♥❣ tü✱ ❣å✐ S ❧➔ t➟♣ ❝→❝ ❞➣② ❈❛✉❝❤② tr♦♥❣ R ù♥❣ ✈ỵ✐ ❧å❝ (Rn)n≥0✳ ❑➼
❤✐➺✉ R = S/∼ = {(an)n≥0 | (an) ∈ S}. ❑❤✐ ✤â (R, +, .) ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥
❝â ✤ì♥ ✈à ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥✿
❱ỵ✐ ♠å✐ (an)n≥0, (bn)n≥0 ∈ R,
(an ) + (bn ) = (an + bn )n≥0 ;
(an ).(bn ) = (an bn )n≥0 .

❚✐➳♣ t❤❡♦✱ ❤❛✐ ♣❤➨♣ t♦→♥ ❝ë♥❣ ✈➔ ♣❤➨♣ ổ ữợ tr M ữủ

ợ ồ (xn), (yn) ∈ M , (xn) + (yn) = (xn + yn)n≥0.
❱ỵ✐ ♠å✐ (an) ∈ R✱ ✈ỵ✐ ♠å✐ (xn) ∈ M ✱ (an).(xn) = (anxn).
❑❤✐ ✤â M ❧➔ R✲♠æ✤✉♥✳
❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R✱ tỉ♣ỉ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ M ❜ð✐ ❧å❝
I ✲❛❞✐❝ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ I ✲❛❞✐❝ ✈➔ ❜❛♦ ✤➛② ✤õ M ✤÷đ❝ ❣å✐ ❧➔ ❜❛♦ ✤➛② ✤õ
I ✲❛❞✐❝✳


✼

✶✳✷ ✣à❛ ♣❤÷ì♥❣ ❤â❛

❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✤à❛ ♣❤÷ì♥❣ ❤â❛ t❤❡♦ ❬✶✹❪✳ ❈❤♦
R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❚➟♣ S ⊂ R ✤÷đ❝ ❣å✐ ❧➔ ♠ët t➟♣ ♥❤➙♥
✤â♥❣ ♥➳✉ 1 ∈ S ✈➔ ✈ỵ✐ ♠å✐ x, y ∈ S t❤➻ xy ∈ S ✳ ❳➨t t➟♣
S × R = {(s, r) | s ∈ S

✈➔ r ∈ R}

✈➔ ✤à♥❤ ♥❣❤➽❛ tr➯♥ S × R ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐✿
∀(s, r), (t, k) ∈ S × R, (s, r) ∼ (t, k) ⇔ ∃u ∈ S : u(ks − tr) = 0.

❑❤✐ ✤â✱ q✉❛♥ ởt q tữỡ ữỡ ợ ộ (s, r) S ì R
t ợ tữỡ ữỡ (s, r) rs t tữỡ (S ì R)/∼ ❧➔ S −1R
❤❛② RS ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤❛✐ t ở ữ s

ợ ồ rs kt ∈ RS ✱ rs + kt = tr +st sk ✈➔ rs . kt = rk
st
❈❤ó♥❣ t❛ ❝â t❤➸ ❦✐➸♠ tr❛ (RS , +, .) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à 11 ✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❱➔♥❤ RS ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❝→❝ t❤÷ì♥❣ ❝õ❛ ✈➔♥❤ R

t÷ì♥❣ ù♥❣ ợ t õ S

ú ỵ r ợ ộ p ∈ ❙♣❡❝(R)✱ S = R \ p ❧➔ ♠ët t➟♣ ♥❤➙♥ ✤â♥❣✳
❑❤✐ ✤â✱ ✈➔♥❤ RS ❝á♥ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ Rp.


✽

▼➺♥❤ ✤➲ ✶✳✷✳✷✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✱ S ❧➔ ♠ët t➟♣
♥❤➙♥ ✤â♥❣ ❝õ❛ R ✈➔ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙②

❧➔ ✤ó♥❣✳
✭✐✮ ❚➟♣ IRS = IS = { rs | r ∈ I ✈➔ s ∈ S} ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ RS ✳
✭✐✐✮ ❱ỵ✐ ♠é✐ p ∈ ❙♣❡❝(R)✱ ❙♣❡❝(Rp) = {qRp | q ∈ ❙♣❡❝(R) ✈➔ q ⊂ p}✳
✭✐✐✐✮ ❱ỵ✐ ♠é✐ p ∈ ❙♣❡❝(R)✱ ✈➔♥❤ Rp ởt ữỡ ợ
ỹ pRp.

M ởt Rổ t tữỡ RS ợ S ❧➔ ♠ët t➟♣
♥❤➙♥ ✤â♥❣✳ ❳➨t t➟♣
S × M = {(s, m) | s ∈ S

✈➔ m ∈ M }✳

❚r➯♥ t➟♣ S × M t❛ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐✿
∀(s, m), (t, n) ∈ S × M, (s, m) ∼ (t, n) ⇔ ∃u ∈ S : u(tm − sn) = 0.

❑❤✐ ✤â✱ q✉❛♥ ❤➺ ∼ ❧➔ ♠ët q✉❛♥ tữỡ ữỡ tr S ì M ợ ộ
m
(s, m) S ì M t ợ tữỡ ữỡ (s, m)
t tữỡ
s
(S ì M )/ ❧➔ S −1 M ❤❛② MS ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ở ổ
ữợ ữ s
ợ ồ ms ✱ nt ∈ MS , ms + nt = tm st+ sn .
❱ỵ✐ ♠å✐ ms ∈ MS ✱ ar ∈ RS ✱ ms . ar = am
.
rs
❈❤ó♥❣ t❛ ❝â t❤➸ ❦✐➸♠ tr❛ MS ❧➔ ♠ët RS ✲♠æ✤✉♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ▼æ✤✉♥ MS tr➯♥ ✈➔♥❤ RS ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ✤à❛
♣❤÷ì♥❣ ❤â❛ ừ M tữỡ ự ợ t õ S

ú ỵ r ợ ộ p (R) S
õ t❛ ❦➼ ❤✐➺✉ MS ❧➔ Mp.

= R\p

❧➔ ♠ët t➟♣ ♥❤➙♥ ✤â♥❣✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✹✳ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ❚➟♣
❙✉♣♣R(M ) = {p ∈ ❙♣❡❝R | Mp = 0}
✤÷đ❝ ❣å✐ ❧➔ ❣✐→ ❝õ❛ ♠ỉ✤✉♥ M ✳

▼➺♥❤ ✤➲ ✶✳✷✳✺✳ ❈❤♦ ❞➣② ❦❤ỵ♣ ♥❣➢♥ ❝→❝ R✲♠æ✤✉♥
0 → N → M → P → 0.

❑❤✐ ✤â ❙✉♣♣R(M ) ⊆ ❙✉♣♣R(N ) ∪ ❙✉♣♣R(P ).

▼➺♥❤ ✤➲ ✶✳✷✳✻✳ ◆➳✉ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ t❤➻
❙✉♣♣R(M ) = V (❆♥♥RM ).

✶✳✸ ❙ü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì
❚r♦♥❣ t✐➳t ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ❝õ❛ ♠ët
♠ỉ✤✉♥ t❤❡♦ ❬✶✺❪✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥ ✈➔ M ❧➔ ♠ët
R✲♠æ✤✉♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❈❤♦ M ❧➔ ♠ët R✲♠æ✤✉♥✳ ❚❛ ❣å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p

❝õ❛ R ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ♥➳✉ tỗ t tỷ x M, x = 0

s ❝❤♦
p = (0 :R x) = ❆♥♥R (x).
❚➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ ❆ssRM ❤♦➦❝
❆ssM ✳
❆ssM = {p ∈ ❙♣❡❝R | p = ❆♥♥(x), x ∈ M }.


✶✵

▼➺♥❤ ✤➲ ✶✳✸✳✷✳ ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✳

✭✐✮ p ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M ❦❤✐ ✈➔ tỗ t ởt ổ
Q ừ M s ❝❤♦ Q ∼
= R/p.
✭✐✐✮ ❈❤♦ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R✲♠æ✤✉♥ ❦❤→❝ ❦❤æ♥❣✳ ●å✐ =
{❆♥♥(x) | x ∈ M }. ❑❤✐ ✤â ♥➳✉ p ❧➔ ♣❤➛♥ tû tè✐ ✤↕✐ ❝õ❛
t❤➻ p ❧➔ ✐✤➯❛♥
♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ M.
✭✐✐✐✮ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ R✲♠æ✤✉♥✳ ❑❤✐ ✤â ❆ssM = ∅ ❦❤✐ ✈➔ ❝❤➾
❦❤✐ M = 0.

▼➺♥❤ ✤➲ ✶✳✸✳✸✳ ●✐↔ sû 0 → M
❝→❝ R✲♠æ✤✉♥✳ ❑❤✐ ✤â ❆ssM

❧➔ ♠ët ❞➣② ❦❤ỵ♣ ♥❣➢♥
∪ ❆ssM .

→M →M →0

⊆ ❆ssM ⊆ ❆ssM

▼➺♥❤ ✤➲ ✶✳✸✳✹✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ởt Rổ
0

ỳ s õ tỗ t↕✐ ♠ët ❞➙② ❝❤✉②➲♥ ❝❤➦t
0 = Mo

M1

...

Mn−1

Mn

t❤ä❛ ♠➣♥ Mi/Mi−1 R/pi✱ ✈ỵ✐ pi ∈ ❙♣❡❝(R)✱ ✈ỵ✐ ♠å✐ i = 1, ..., n. ❍ì♥
♥ú❛ ❆ssR(M ) ⊆ {p1, ..., pn} ⊂ ❙✉♣♣R(M ) ✈➔ t➟♣ ❝→❝ ♣❤➛♥ tû ❝ü❝ t✐➸✉ ❝õ❛
✸ t➟♣ ♥➔② ❧➔ trị♥❣ ♥❤❛✉✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✺✳ ▼ët R✲♠ỉ✤✉♥ ✤÷đ❝ ❣å✐ ❧➔ ✤è✐ ♥❣✉②➯♥ sì ♥➳✉ ♥â ❝â

❞✉② ♥❤➜t ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t✳ ▼ët ♠æ✤✉♥ ❝♦♥ N ❝õ❛ M ✤÷đ❝
❣å✐ ❧➔ ♠ët ♠ỉ✤✉♥ ❝♦♥ ♥❣✉②➯♥ sì ❝õ❛ M ♥➳✉ M/N ❧➔ ✤è✐ ♥❣✉②➯♥ sì✳ ◆➳✉
❆ss(M/N ) = {p}✱ t❛ ♥â✐ N ❧➔ p✲♥❣✉②➯♥ sì ❤❛② N ❧✐➯♥ ❦➳t ✈ỵ✐ p✳


✶✶
❈❤♦ N ❧➔ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ▼ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì ❝õ❛ N ❧➔ ♠ët
❜✐➸✉ ❞✐➵♥ N = M1 ∩ M2 ∩ ... ∩ Mn tr♦♥❣ ✤â Mi ❧➔ ❝→❝ ♠ỉ✤✉♥ ❝♦♥ pi✲♥❣✉②➯♥
sì ❝õ❛ M. P❤➙♥ t➼❝❤ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ t❤✉ ❣å♥ ♥➳✉ ❝→❝ pi ❧➔ ✤ỉ✐ ởt

t ổ õ Mi tứ

ú ỵ ✭✐✮ ◆➳✉ M1 ✈➔ M2 ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ p✲♥❣✉②➯♥ sì ❝õ❛ M

t❤➻ M1 ∩ M2 ❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ p✲♥❣✉②➯♥ sì ❝õ❛ M ✳ ❱➻ t❤➳ ♠å✐ ♣❤➙♥ t➼❝❤
♥❣✉②➯♥ sì ❝õ❛ ♠ỉ✤✉♥ ❝♦♥ N ✤➲✉ ❝â t❤➸ q✉② ✈➲ ♠ët ♣❤➙♥ t➼❝❤ t❤✉ ❣å♥✳
✭✐✐✮ ❑❤✐ M = R ✈➔ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r t❤➻ ❦❤→✐ ♥✐➺♠ ✐✤➯❛♥ ♥❣✉②➯♥ sì
trị♥❣ ợ ổ sỡ

ỵ ồ ♠æ✤✉♥ ❝♦♥ t❤ü❝ sü ❝õ❛ ♠ët ♠æ✤✉♥ ◆♦❡t❤❡r ✤➲✉ ❝â
sü ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì rót ❣å♥✳

❇ê ✤➲ ✶✳✸✳✽✳ ◆➳✉ N = Q1 ∩ ... ∩ Qr ❧➔ ♠ët ♣❤➙♥ t➼❝❤ ♥❣✉②➯♥ sì rót ❣å♥

✈➔ Qi ❧✐➯♥ ❦➳t ✈ỵ✐ pi t❤➻ t õ

ss(M/N ) = {p1, ..., pr }.

ỵ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ ♠ët
R✲♠ỉ✤✉♥

❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â 0 =

❝♦♥ p✲♥❣✉②➯♥ sì✳

p∈❆ssM

Q(p)✱

tr♦♥❣ ✤â Q(p) ❧➔ ♠ỉ✤✉♥

✶✳✹ ❈❤✐➲✉ ❑r✉❧❧
❚r♦♥❣ t✐➳t ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ❑r✉❧❧ ✈➔ ♠ët sè ❦➳t
q✉↔ t❤❡♦ ❬✶✹❪ ✈➔ ❬✶✺❪✳


✶✷
❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à 1 = 0 ởt ỳ ỗ
n + 1 ✐✤➯❛♥ ♥❣✉②➯♥ tè p0 ⊃ p1 ⊃ . . . ⊃ pn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➙② ❝❤✉②➲♥
♥❣✉②➯♥ tè ✤ë ❞➔✐ n✳ ◆➳✉ p ∈ ❙♣❡❝R✱ ❝❤➦♥ tr➯♥ ♥❤ä ♥❤➜t ❝õ❛ t➜t ❝↔ ✤ë ❞➔✐
❝õ❛ ❝→❝ ❞➙② ❝❤✉②➲♥ ♥❣✉②➯♥ tè ợ p = p0 ữủ ồ ở ừ p ✈➔ ❦➼
❤✐➺✉ ❧➔ ❤tp✳
❤t(p) = s✉♣{n | p0 = p ⊃ p1 ⊃ . . . ⊃ pn}.
❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ t❤ü❝ sü ❝õ❛ R✳ ❈❤ó♥❣ t❛ ✤à♥❤ ở ừ I
ữợ ợ t ừ ❝→❝ ✤ë ❝❛♦ ❝õ❛ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝❤ù❛ I ✿
❤tI = inf{❤tp | p ∈ ❙♣❡❝(R) ✈➔ p ⊇ I}.

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ❈❤✐➲✉ ❝õ❛ R ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❝❤➦♥ tr➯♥ ♥❤ä ♥❤➜t

❝õ❛ t➜t ❝↔ ✤ë ❝❛♦ ❝õ❛ t➜t ❝↔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✿

dim R = sup{❤tp | p ∈ ❙♣❡❝R}.

◆â ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ R✳

❱➼ ❞ư ✶✳✹✳✷✳ 1) ❈❤♦ K ❧➔ ♠ët tr÷í♥❣✳ ❑❤✐ ✤â dim K = 0✳
2) dim(Z) = 1✳

◆❤➟♥ ①➨t ✶✳✹✳✸✳ (i) ❱ỵ✐ ♠é✐ p ∈ ❙♣❡❝R, ❤t(p) = dim(Rp).
(ii)

❱ỵ✐ ♠é✐ ✐✤➯❛♥ I ❝õ❛ R✱ dim(R/I) + ❤t(I)

dim R.

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✹✳ ❈❤♦ M = 0 ❧➔ ♠ët R✲♠æ✤✉♥✳ ❈❤✐➲✉ ❑r✉❧❧ ❝õ❛ M ❧➔
dim(M ) = dim(R/❆♥♥(M )).

◆➳✉ M = 0✱ q✉✐ ÷ỵ❝ dim(M ) = −1.


✶✸

▼➺♥❤ ✤➲ ✶✳✹✳✺✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✈➔ M = 0 ❧➔

♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳
(i) M ❧➔ ♠ët R✲♠æ✤✉♥ ❝â ✤ë ❞➔✐ ❤ú✉ ❤↕♥✳
(ii) ❱➔♥❤ R/M rt
(iii) dim M = 0.

ỵ ❈❤♦ 0 → M
R✲♠ỉ✤✉♥✳

→M →M →0

❧➔ ♠ët ❞➣② ❦❤ỵ♣ ♥❣➢♥

õ

M = {M , M

}.

ỵ rt (R, m) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r
✤à❛ ♣❤÷ì♥❣ ✈➔ M = 0 ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â
❧➔ ♠ët ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè ❤ú✉ t➾ ❦❤✐ n 0 ✈➔

❞✐♠(M ) = ❞❡❣ R(M/mnM )
= ✐♥❢{r ∈ N | ∃x1 , ..., xr ∈ m s❛♦ ❝❤♦
❈❤ó þ r➡♥❣✱ ♥➳✉ d

n
R (M/m M )

(M/(x1 , . . . , xr )M ) < ∞}.

✈➔ ❤➺ ♣❤➛♥ tû x1, . . . , xd ∈ m s❛♦ ❝❤♦
(M/(x1 , . . . , xd )M ) < ∞ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➺ t❤❛♠ sè ❝õ❛ M ✳
= dim M

▼➺♥❤ ✤➲ ✶✳✹✳✽✳ ❈❤♦ (R, m) ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣

✈➔ M = 0 ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â
(i) dimR (M ) = dimR (M ).
(ii) dim(M ) = max{dim(R/p) | p ∈ ❆ssM }.


✶✹

✶✳✺ ▼æ✤✉♥ ❆rt✐♥

❈❤♦ m ❧➔ ♠ët ✐✤➯❛♥ ❝ü❝ ✤↕✐ ❝õ❛ ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥ R✳ ◆❤➢❝ ❧↕✐
r➡♥❣ ♠æ✤✉♥ ❝♦♥ m✲①♦➢♥ Γm(A) ❝õ❛ A ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐
(0 :A mn )

Γm (A) =

n≥0

❑❤✐ ✤â✱ t❛ ❝â ❦➳t q✉↔ s❛✉✳

▼➺♥❤ ✤➲ ✶✳✺✳✶✳ ✭❳❡♠ ❬✷✵❪✱ ▼➺♥❤ ✤➲ ✶✳✹✱ ❇ê ✤➲ ✶✳✻✮✳

✭✐✮ ●✐↔ sû A ❧➔ ♠ët R✲♠æ✤✉♥ ❆rt✐♥ ❦❤→❝ ❦❤æ♥❣✳ ❑❤✐ ✤â ❝❤➾ ❝â ❤ú✉ ❤↕♥
✐✤➯❛♥ ❝ü❝ ✤↕✐ m ❝õ❛ R s❛♦ ❝❤♦ Γm(A) = 0✳ ◆➳✉ ❝→❝ ✐✤➯❛♥ ❝ü❝ ✤↕✐ ♣❤➙♥
❜✐➺t ✤â ❧➔ m1, m2, ..., mr t❤➻
A = Γm1 (A) ⊕ ... ⊕ Γmr (A)

✈➔ ❙✉♣♣A = {m1, ..., mr }.

✭✐✐✮ ❱ỵ✐ ♠é✐ j ∈ {1, ..., r}✱ ♥➳✉ s ∈ R\mj , t❤➻ ♣❤➨♣ ♥❤➙♥ ❜ð✐ s ❝❤♦ t❛
♠ët tü ✤➥♥❣ ❝➜✉ ❝õ❛ Γm (A). ❉♦ ✤â Γm (A) ❝â ❝➜✉ tró❝ tü ♥❤✐➯♥ ❝õ❛ ♠ët
Rm ✲♠ỉ✤✉♥ ✈➔ ✈ỵ✐ ❝➜✉ tró❝ ♥➔②✱ ♠ët t➟♣ ❝♦♥ ❝õ❛ Γm (A) ❧➔ ♠ët R✲♠æ✤✉♥
❝♦♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ Rm ✲♠æ✤✉♥ ❝♦♥✳ ✣➦❝ ❜✐➺t
j

j

j

j

j

Amj ∼
= Γmj (A),

✈ỵ✐ ♠å✐ j = 1, ..., r.

❑➼ ❤✐➺✉ ✶✳✺✳✷✳ ✣➸ ❝❤♦ t❤✉➟♥ t✐➺♥✱ tø ❣✐í trð ✤✐ t❛ ✤➦t
A = A1 ⊕ ... ⊕ Ar

✈➔ JA =

m,
m∈❙✉♣♣A

tr♦♥❣ ✤â Aj =

(0 :A mnj ) (1
n>0

✤à❛ ♣❤÷ì♥❣ t❤➻ JA = m.

j

r).

ú ỵ r (R, m)

✶✺

▼➺♥❤ ✤➲ ✶✳✺✳✸✳ ✭❳❡♠ ❬✷✵❪✱ ❇ê ✤➲ ✶✳✶✱ ❍➺ q✉↔ ✶✳✶✷✮✳ ❈❤♦ A ❧➔ R✲♠æ✤✉♥

❆rt✐♥ ❦❤→❝ ❦❤æ♥❣ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m)✳ ❑❤✐ ✤â✱ A ❝â ❝➜✉ tró❝ tü
♥❤✐➯♥ ❝õ❛ R✲♠æ✤✉♥✱ tr♦♥❣ ✤â R ❧➔ ✈➔♥❤ ✤➛② ✤õ t❤❡♦ tæ♣æ m✲❛❞✐❝ ❝õ❛ R ✈➔
♠å✐ t➟♣ ❝♦♥ ❝õ❛ A ❧➔ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ A ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ R✲♠ỉ✤✉♥
❝♦♥ ❝õ❛ A✳ ❉♦ ✤â✱ A ❝â ❝➜✉ tró❝ tü ♥❤✐➯♥ ❝õ❛ R✲♠ỉ✤✉♥ ❆rt✐♥✳

❈❤♦ (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✤➛② ✤õ✳ ❑➼ ❤✐➺✉ E(R/m) ❧➔ ❜❛♦ ♥ë✐ ①↕
❝õ❛ tr÷í♥❣ t❤➦♥❣ ❞÷ R/m ❝õ❛ R✳ ❳➨t ❤➔♠ tû D(−) = ❍♦♠R(−, E(R/m))
tø ♣❤↕♠ trị ❝→❝ R✲♠ỉ✤✉♥ ✤➳♥ ❝❤➼♥❤ ♥â✳ ❱➻ E(R/m) ❧➔ ♠ỉ✤✉♥ ♥ë✐ ①↕ ♥➯♥
D(−) ❧➔ ❤➔♠ tû ❦❤ỵ♣✳ ❚❛ ❣å✐ D(−) ❧➔ ❤➔♠ tû ✤è✐ ♥❣➝✉ ▼❛t❧✐s✳
❑❤✐ ✤â t❛ ❝â ❦➳t q✉↔ s❛✉ ❝õ❛ ❊✳ ▼❛t❧✐s ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✭❬✶✻❪✱ ✣à♥❤
❧➼ ✹✳✷✮✳

▼➺♥❤ ✤➲ ✶✳✺✳✹✳ ✭✐✮ R✲♠æ✤✉♥ E rt ợ ộ f R(E, E), tỗ

t ♥❤➜t af ∈ R : f (x) = af x, ✈ỵ✐ ♠å✐ x ∈ E.
✭✐✐✮ ◆➳✉ N ❧➔ R✲♠ỉ✤✉♥ ◆♦❡t❤❡r✱ t❤➻ D(N ) ❧➔ ❆rt✐♥✳
✭✐✐✐✮ ◆➳✉ A ❧➔ R✲♠æ✤✉♥ ❆rt✐♥✱ t❤➻ D(A) ❧➔ ◆♦❡t❤❡r✳
✭✐✈✮ ❆♥♥M = ❆♥♥D(M ), ✈➔ ♥➳✉ M ❧➔ R✲♠æ✤✉♥ s❛♦ ❝❤♦
R (D(M ))

=

R (M )

< ∞,

t❤➻

R (M ).

✶✳✻ ❇✐➸✉ ❞✐➵♥ t❤ù ❝➜♣
❑❤→✐ ♥✐➺♠ ♣❤➙♥ t➼❝❤ ✤è✐ ♥❣✉②➯♥ sì ❝❤♦ ♠ỉ✤✉♥ ❆rt✐♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉
❜ð✐ ❉✳ ❑✐r❜② ❬✶✵❪ ✈➔ s❛✉ ✤â ■✳ ●✳ ▼❛❝❞♦♥❛❧❞ ❬✶✷❪ tr➻♥❤ ❜➔② ởt tờ
qt ổ tũ ỵ ổ ồ ❧➔ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣✳ ❈❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔②
t❤✉➟t ♥❣ú ❝õ❛ ▼❛❝❞♦♥❛❧❞ ❬✶✷❪✳ ❑➼ ❤✐➺✉ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥✳


✶✻

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳✶✳ ✭✐✮ ▼ët ❘✲♠ỉ✤✉♥ ▲ ✤÷đ❝ ❣å✐ ❧➔ t❤ù ❝➜♣ ♥➳✉ ▲ = 0 ✈➔

✈ỵ✐ ♠é✐ ① ∈ R, ♣❤➨♣ ♥❤➙♥ ❜ð✐ ① tr➯♥ ▲ ❧➔ t♦➔♥ ❝➜✉ ❤♦➦❝ ❧ơ② ❧✐♥❤✳ ❚r♦♥❣
tr÷í♥❣ ❤đ♣ ♥➔②✱ t➟♣ ❤đ♣ ❝→❝ ♣❤➛♥ tû ① ∈ R s❛♦ ❝❤♦ ♣❤➨♣ ♥❤➙♥ ❜ð✐ ① tr➯♥
▲ ❧➔ ❧ô② ❧✐♥❤ ❧➔♠ t❤➔♥❤ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✱ ❝❤➥♥❣ ❤↕♥ ❧➔ p✱ ✈➔ t❛ ❣å✐ ▲
❧➔ p✲t❤ù ❝➜♣✳
✭✐✐✮ ❈❤♦ ▲ ❧➔ ❘✲♠æ✤✉♥✳ ▼ët ❜✐➸✉ ❞✐➵♥ ▲1 ✰✳✳✳✰ ▲♥✱ tr♦♥❣ ✤â ♠é✐ ▲✐ ❧➔
♠ỉ✤✉♥ ❝♦♥ p✐✲t❤ù ❝➜♣ ▲✱ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ▲✳ ◆➳✉
▲❂✵ ❤♦➦❝ ▲ ❝â ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ t❤➻ t❛ ♥â✐ ▲ ❧➔ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝✳ ❇✐➸✉ ❞✐➵♥
♥➔② ✤÷đ❝ ❣å✐ ❧➔ tè✐ t✐➸✉ ♥➳✉ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè p✐ ❧➔ ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉
✈➔ ♠é✐ ▲✐ ❧➔ ❦❤ỉ♥❣ t❤ø❛ ✈ỵ✐ ♠å✐ = 1, .., .
ú ỵ r 1 2 ❧➔ ❝→❝ ♠æ✤✉♥ ❝♦♥ p t❤ù ❝➜♣ ❝õ❛ ▲ t❤➻ ▲1 ✰ ▲2
❝ơ♥❣ ❧➔ ♠ỉ✤✉♥ ❝♦♥ p✲t❤ù ❝➜♣ ❝õ❛ ▲✳ ❱➻ t❤➳ ♠å✐ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ ❝õ❛ ▲
✤➲✉ ❝â t❤➸ ✤÷❛ ✤÷đ❝ ✈➲ ❞↕♥❣ tè✐ t✐➸✉ ❜➡♥❣ ❝→❝❤ ❜ä ✤✐ ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥
t❤ø❛ ✈➔ ❣ë♣ ❧↕✐ ♥❤ú♥❣ t❤➔♥❤ ♣❤➛♥ ❝ị♥❣ ❝❤✉♥❣ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè✳ ❚➟♣
❤đ♣ {p1, ..., p♥} ❧➔ ✤ë❝ ❧➟♣ ✈ỵ✐ ✈✐➺❝ ❝❤å♥ ❜✐➸✉ ❞✐➵♥ t❤ù ❝➜♣ tè✐ t✐➸✉ ❝õ❛ ▲
✈➔ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥ ❦➳t ❝õ❛ ▲✱ ❦➼ ❤✐➺✉ ❧➔ ❆ttR▲.

❈→❝ tỷ ợ = 1, ..., ữủ ❣å✐ ❧➔ ❝→❝ t❤➔♥❤ ♣❤➛♥ t❤ù ❝➜♣ ❝õ❛ ▲✳
◆➳✉ p✐ ❧➔ tè✐ t✐➸✉ tr♦♥❣ t➟♣ ❆ttR▲ t❤➻ p✐ ✤÷đ❝ ❣å✐ ❧➔ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❣➢♥
❦➳t ❝æ ❧➟♣ ❝õ❛ ▲ ✈➔ ▲✐ ✤÷đ❝ ❣å✐ ❧➔ t❤➔♥❤ ♣❤➛♥ t❤ù ❝➜♣ ❝ỉ ❧➟♣ ừ

ỵ ồ Rổ rt A ❞✐➵♥ ✤÷đ❝✳


✶✼

▼➺♥❤ ✤➲ ✶✳✻✳✸✳ ●✐↔ sû ▲ ❧➔ ♠ët ❘✲♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥ ✤÷đ❝✳ ❑❤✐ ✤â ❝→❝

♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✿
✭✐✮ ❆ttR▲ = ∅ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ▲ = 0.
✭✐✐✮ ♠✐♥❆ttR▲ = ♠✐♥❱❛r(❆♥♥R▲). ✣➦❝ ❜✐➺t✱

❞✐♠(R/❆♥♥R▲) = ♠❛①{❞✐♠(R/p)| p ∈ ❆ttR▲}.
✭✐✐✐✮ ❈❤♦ 0 L
ữủ õ t õ





ttR

0

ợ ❘✲♠æ✤✉♥ ❜✐➸✉ ❞✐➵♥

⊆ ❆ttR ▲ ⊆ ❆ttR ▲✬ ∪ ❆ttR ▲ .

▼➺♥❤ ✤➲ ✶✳✻✳✹✳ ✭❳❡♠ ❬✷❪✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ❣✐❛♦ ❤♦→♥ ✤à❛
♣❤÷ì♥❣ ✈➔ A ❧➔ ♠ët R✲♠ỉ✤✉♥ ❆rt✐♥✳ ❑❤✐ ✤â

❆ttR❆ = {p ∩ R | p ∈ ❆ttR❆}.

✶✳✼ ❈❤✐➲✉ ◆♦❡t❤❡r
❚r♦♥❣ ❬✶✾❪✱ ❘✳ ◆✳ ❘♦❜❡rts ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ❑r✉❧❧ ✭❦➼ ❤✐➺✉ ❧➔
❑❞✐♠✮ ❝❤♦ ♠ỉ✤✉♥ tị② þ ✈➔ ✤÷❛ r❛ ♠ët sè ❦➳t q✉↔ ✈➲ ❝❤✐➲✉ ❑r✉❧❧ ♥➔② ❝❤♦
❝→❝ ♠ỉ✤✉♥ ❆rt✐♥✳ ✣➸ tr→♥❤ ♥❤➛♠ ❧➝♥ ✈ỵ✐ ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ❑r✉❧❧ ❝õ❛ ❝→❝
♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✱ ❉✳ ❑✐r❜② tr♦♥❣ ❬✶✵❪ ✤➣ ✤ê✐ t❤✉➟t ♥❣ú ❝õ❛ ❘♦❜❡rts
✈➔ ✤➲ ①✉➜t t❤➔♥❤ ❝❤✐➲✉ ◆♦❡t❤❡r✳ ❙❛✉ ✤➙② ❧➔ ❦❤→✐ ♥✐➺♠ ❝❤✐➲✉ ◆♦❡t❤❡r ❝❤♦
♠æ✤✉♥ ❆rt✐♥ t❤❡♦ t❤✉➟t ♥❣ú ❝õ❛ ❉✳ ❑✐r❜② ❬✶✵❪✳ ❑➼ ❤✐➺✉ R ❧➔ ✈➔♥❤ ◆♦❡t❤❡r
❣✐❛♦ ❤♦→♥✳


✶✽

✣à♥❤ ♥❣❤➽❛ ✶✳✼✳✶✳ ❈❤✐➲✉ ◆♦❡t❤❡r ❝õ❛ ❘✲♠æ✤✉♥ ❆rt✐♥ ❆✱ ❦➼ ❤✐➺✉ ❜ð✐
◆✲❞✐♠R❆, ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿
❑❤✐ ❆ = 0, ✤➦t ◆✲❞✐♠R❆ = −1✳
❱ỵ✐ A = 0✱ ❝❤♦ ♠ët sè ♥❣✉②➯♥ d ≥ 0, t❛ ✤➦t ◆✲❞✐♠R❆ = d ♥➳✉
◆✲❞✐♠R❆ < d ❧➔ s❛✐ ✈➔ ✈ỵ✐ ♠é✐ ❞➣② t➠♥❣ ❝→❝ ổ 0 1 ...
ừ tỗ t ♠ët sè tü ♥❤✐➯♥ ♥0 s❛♦ ❝❤♦ ◆✲❞✐♠R(❆♥+1/❆♥) < d ✈ỵ✐ ♠å✐
♥ > ♥0.

❱➼ ❞ư ✶✳✼✳✷✳ ❈❤♦ M ❧➔ R✲♠ỉ✤✉♥ ❦❤→❝ ❦❤æ♥❣✳ ❑❤✐ ✤â M ❧➔ R✲♠æ✤✉♥

◆♦❡t❤❡r ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ◆✲dimR M = 0✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû M ❧➔ R✲♠æ✤✉♥
◆♦❡t❤❡r✳ ❱➻ ♠å✐ ❞➣② t➠♥❣ M0 ⊆ M1 ⊆ ... ⊆ Mn ⊆ ... ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛

M ✤➲✉ ứ tỗ t n0 N s Mn = Mn+1 , ✈ỵ✐ ♠å✐ n > n0 ✳ ❉♦
✤â✱ Mn+1/Mn = 0✱ ✈➻ t❤➳ ◆✲dimR Mn+1/Mn = −1 < 0✱ ✈ỵ✐ ♠å✐ n > n0✳
❱➻ M = 0 ♥➯♥ ◆✲dimR M ≥ 0 ✈➔ ❞♦ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ◆✲ dimR M = 0✳
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ◆✲dimR M = 0✳ ❑❤✐ ✤â ❧➜② ♠ët ❞➣② t➠♥❣ ❜➜t ❦ý N0 ⊆
N1 ⊆ ... ⊆ ... ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M tỗ t số
ữỡ n0 s❛♦ ❝❤♦ ◆✲dim Nk+1/Nk = −1 < 0, ✈ỵ✐ ♠å✐ k > n0. ❉♦ ✤â
Nk+1 = Nk ✈ỵ✐ ♠å✐ n > n0 ❤❛② ❞➣② tr➯♥ ❧➔ ❞ø♥❣✱ ♥❣❤➽❛ ❧➔ M Rổ
tr

ỵ ỵ
ỳ t ❦❤✐ n

0

(0 :A JAn )

❧➔ ♠ët ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè

✈➔

◆✲❞✐♠RA = ❞❡❣( R(0 :A JAn ))
= ✐♥❢{t | ∃x1 , ..., xt ∈ JA |

R (0 :A

(x1 , ..., xt )R) < ∞}.

●✐↔ sû d = ◆✲❞✐♠A t❤➻ ❤➺ x = (x1, ..., xd) ❝→❝ ♣❤➛♥ tû tr♦♥❣ JA s❛♦
❝❤♦ R(0 :A xR) < ∞ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ t❤❛♠ sè ❝õ❛ A✳

✶✾

▼➺♥❤ ✤➲ ✶✳✼✳✹✳ ✭❳❡♠ ❬✺❪✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ A ❧➔ R✲♠ỉ✤✉♥

❆rt✐♥✳ ❑❤✐ ✤â✱ A ❝â ❝➜✉ tró❝ tü ♥❤✐➯♥ ❝õ❛ R✲♠ỉ✤✉♥ ❆rt✐♥ ✈➔ t❛ ❝â

◆✲❞✐♠RA = ◆✲❞✐♠RA.
❈❤➼♥❤ ✈➻ ✈➟②✱ t❛ ❝â t❤➸ ✈✐➳t ◆✲❞✐♠A t❤❛② ❝❤♦ ◆✲❞✐♠RA ❤♦➦❝ ◆✲❞✐♠RA.

▼➺♥❤ ✤➲ ✶✳✼✳✺✳ ✭❳❡♠ ❬✺❪✮ ✭✐✮ ◆✲❞✐♠A = 0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ A = 0 ✈➔
R (A)

< ∞✳

❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ❆ttRA = {m}. ❍ì♥ ♥ú❛✱ ♥➳✉
0 −→ A −→ A −→ A −→ 0

❧➔ ❞➣② ❦❤ỵ♣ ❝→❝ R✲♠ỉ✤✉♥ ❆rt✐♥ t❤➻

◆✲❞✐♠RA = ♠❛①{◆✲❞✐♠RA , ◆✲❞✐♠RA }.
✭✐✐✮ ◆✲❞✐♠A ≤ dim R/❆♥♥RA = ♠❛①{❞✐♠R/p | p ∈ ❆ttR(A)}.
✭✐✐✐✮ ◆✲❞✐♠A = ❞✐♠R/❆♥♥RA = ♠❛①{❞✐♠R/p | p ∈ ❆ttR(A)}.

✶✳✽ ▼æ✤✉♥ ố ỗ ữỡ
ỵ tt ố ỗ ữỡ ữủ ợ t t
rt ỳ ♥➠♠ ✶✾✻✵✱ s❛✉ ✤â ✤÷đ❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐
r➜t t ồ tr t ợ ữ rtsr r
t ỵ tt ố ỗ ữỡ õ ỳ
ự ử t ợ tr ỹ ừ ồ rữợ t ú tổ

tr ❦❤→✐ ♥✐➺♠ ❤➔♠ tû ①♦➢♥ t❤❡♦ ❬✷❪✳
❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r✱ M ❧➔ R✲♠æ✤✉♥ ✈➔ I ⊂ R ❧➔ ♠ët
✐✤➯❛♥ ❝õ❛ R✳ ❑❤✐ ✤â t❛ ❝â ❤➔♠ tû I ✲①♦➢♥ ΓI (−) tø ♣❤↕♠ trị ❝→❝ R✲♠ỉ✤✉♥


✷✵
✈➔♦ ❝❤➼♥❤ ♥â ❧➔ ❤✐➺♣ ❜✐➳♥✱ ❝ë♥❣ t➼♥❤ ✈➔ ❦❤ỵ♣ tr ợ ộ Rổ M,
I (M ) ữủ ❜ð✐ ❝ỉ♥❣ t❤ù❝
∞

(0 :M I n ).

ΓI (M ) =

n=0

❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ i ❦❤æ♥❣ ➙♠✱ t❛ ❝â ❤➔♠ tû ❞➝♥ ①✉➜t ♣❤↔✐ t❤ù i RiΓI (−)
❝õ❛ ❤➔♠ tû ΓI () õ ổ ố ỗ ữỡ HIi (M ) tự
i ừ Rổ M ợ I ữủ ①→❝ ✤à♥❤ ❜ð✐
HIi (M ) = Ri ΓI (M ).

✣à♥❤ ỵ M Rổ õ HIi (M ) = 0✱ ✈ỵ✐ ♠å✐
i > ❞✐♠M ✳

✭✐✐✮ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝
❦❤æ♥❣ ✈➔ ❝❤✐➲✉ ❑r✉❧❧ ❞✐♠M = d õ Hmd (M ) = 0

ỵ ✭✐✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ❧➔ R✲♠æ✤✉♥

❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â✱ R✲♠æ✤✉♥ Hmi (M ) ❧➔ ❆rt✐♥ ✈ỵ✐ ♠å✐ i ∈ N0.

✭✐✐✮ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ I ❧➔ ♠ët ✐✤➯❛♥ ❜➜t ❦➻ ❝õ❛ R✱
M ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❦❤→❝ ❦❤æ♥❣ ❝â ❝❤✐➲✉ ❑r✉❧❧ ❞✐♠M = d✳ ❑❤✐
✤â✱ R✲♠æ✤✉♥ HId(M ) ❧➔ ❆rt✐♥✳

▼➺♥❤ ✤➲ ✶✳✽✳✸✳ ✭❳❡♠ ❬✷❪✮ ❈❤♦ (R, m) ❧➔ tr ữỡ M
ỳ s ợ M = d✳ ❑❤✐ ✤â

❆ttR(Hmd (M )) = {p ∈ ❆ssRM | R/p = d}.

ỵ ợ ♥❣➢♥ ❝→❝ R✲♠æ✤✉♥ 0 → M
M → M → 0✳

→

❑❤✐ õ t õ ởt ợ ổ ố ỗ ✤✐➲✉ ✤à❛

♣❤÷ì♥❣
0 → Hm0 (M ) → Hm0 (M ) → Hm0 (M ) → Hm1 (M ) → ...


✷✶

▼➺♥❤ ✤➲ ✶✳✽✳✺✳ ✭❳❡♠ ❬✷❪✮ ❈❤♦ p ∈ ❆ssR(M ) ✈ỵ✐ ❞✐♠R/p = t✳ ❑❤✐ ✤â
Hmt (M ) = 0

✈➔ p ∈ ❆ttRHmt (M ).

✶✳✾ ❚➼♥❤ ❝❛t❡♥❛r② ❝õ❛ ✈➔♥❤
❚r♦♥❣ t✐➳t ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ✈➲ t➼♥❤
❝❛t❡♥❛r② ❝õ❛ ✈➔♥❤ t❤❡♦ ❬✶✹❪✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳✶✳ ❈❤♦ q ⊂ p ❧➔ ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ▼ët ❞➣② ❝→❝

✐✤➯❛♥ ♥❣✉②➯♥ tè q = p0 ⊂ p1 ⊂ . . . ⊂ pn = p s❛♦ ❝❤♦ pi = pi+1✱ ✈ỵ✐ ♠å✐
i = 0, . . . , n − 1 ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➣② ❝→❝ ✐✤➯❛♥ ♥❣✉②➯♥ tè ❜➣♦ ❤á❛ ❣✐ú❛ q
✈➔ p ♥➳✉ ợ ồ 0 i n 1 ổ tỗ t ✐✤➯❛♥ ♥❣✉②➯♥ tè P ♥➔♦ t❤ä❛
♠➣♥ pi ⊂ P ⊂ pi+1 ✈➔ pi = P = pi+1✳ ❑❤✐ ✤â n ✤÷đ❝ ❣å✐ ❧➔ ✤ë ❞➔✐ ❝õ❛ ❞➣②
✐✤➯❛♥ ♥❣✉②➯♥ tè ❜➣♦ ❤á❛ tr➯♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳✷✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✈➔♥❤ ❝❛t❡♥❛r② ♥➳✉ ✈ỵ✐ ♠å✐ ❝➦♣

✐✤➯❛♥ ♥❣✉②➯♥ tè q ⊂ p ừ R ổ tỗ t ởt tố ❜➣♦
❤á❛ ❣✐ú❛ q ✈➔ p✱ ✈➔ ♠å✐ ❞➣② ♥❣✉②➯♥ tè ❜➣♦ ❤á❛ ❣✐ú❛ q ✈➔ p ✤➲✉ ❝â ❝ị♥❣ ✤ë
❞➔✐✳
▲ỵ♣ ✈➔♥❤ ❝❛t❡♥❛r② ✤➛✉ t✐➯♥ ✤÷đ❝ ❝❤➾ r❛ ❜ð✐ ❲✳ ❑r✉❧❧ ♥➠♠ ✶✾✸✼✳ ➷♥❣
❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ K ❧➔ ♠ët tr÷í♥❣ t❤➻ ♠å✐ K ✲✤↕✐ sè ❤ú✉ ❤↕♥ s✐♥❤ ✤➲✉
❧➔ ✈➔♥❤ ❝❛t❡♥❛r②✳ ◆➠♠ ✶✾✹✻✱ ■✳ ❈♦❤❡♥ ❬✸❪ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♠å✐ ✈➔♥❤ ✤à❛
♣❤÷ì♥❣ ✤➛② ✤õ ❧➔ ❝❛t❡♥❛r②✳ ❙❛✉ ✤â✱ ▼✳ ◆❛❣❛t❛ ❬✶✼❪ ✤➣ ❝❤ù♥❣ tä r➡♥❣ ♠å✐
♠✐➲♥ ♥❣✉②➯♥✱ ✤à❛ ♣❤÷ì♥❣ tü❛ ❦❤æ♥❣ trë♥ ❧➝♥ ❧➔ ❝❛t❡♥❛r②✳ ◆➳✉ R ❧➔ ✈➔♥❤
❝❛t❡♥❛r② t❤➻ Rp ❧➔ ❝❛t❡♥❛r② ✈ỵ✐ ♠å✐ p ∈ ❙♣❡❝R✳ ❍ì♥ ♥ú❛✱ ✈➔♥❤ t❤÷ì♥❣ ❝õ❛
✈➔♥❤ ❝❛t❡♥❛r② ❧➔ ❝❛t❡♥❛r②✳ ❱➻ t❤➳ ❤➛✉ ❤➳t ❝→❝ ✈➔♥❤ ✤÷đ❝ ❜✐➳t ✤➳♥ tr♦♥❣


✷✷
❍➻♥❤ ❤å❝ ✤↕✐ sè ✤➲✉ ❧➔ ❝❛t❡♥❛r②✳

▼➺♥❤ ✤➲ ✶✳✾✳✸✳ ✭❳❡♠ ❬✶✽❪✮ ❈❤♦ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✳
❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿
(i) R ❧➔ ❝❛t❡♥❛r②✳
(ii) dim R/q = dim R/p + ❤tp/q ✈ỵ✐ ♠å✐ q ⊆ p; p, q ∈ ❙♣❡❝R.

◆❤➢❝ ❧↕✐ r➡♥❣✱ ✈➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ✤➥♥❣ ❝❤✐➲✉ ♥➳✉ dim R/p = dim R
✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè tè✐ t❤✐➸✉ p ❝õ❛ R✳ ❱ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛
R t❛ ❧✉ỉ♥ ❝â ❜➜t ✤➥♥❣ t❤ù❝
❤tp + dim R/p

dim R.

◆➠♠ ✶✾✼✶✱ ❘✳ ❏✳ ❘❛t❧✐❢❢ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ❝→❝ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✳

▼➺♥❤ ✤➲ ✶✳✾✳✹✳ ✭❳❡♠ ❬✶✽❪✮ ●✐↔ sû R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r ✤➥♥❣

❝❤✐➲✉✳ ❑❤✐ ✤â R ❧➔ ❝❛t❡♥❛r② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛
R t❛ ❝â

❤tp + dim R/p = dim R✳

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳✺✳ ✭❳❡♠ ❬✶✼❪✮ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ tü❛ ❦❤ỉ♥❣ trë♥ ❧➝♥ ♥➳✉

✈➔♥❤ ✤➛② ✤õ m✲❛❞✐❝ R ❝õ❛ R ❧➔ ✤➥♥❣ ❝❤✐➲✉✱ tù❝ ❧➔ dim R/p = dim R ✈ỵ✐ ♠å✐
p ∈ min ❆ss(R)✳ ❱➔♥❤ R ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ trë♥ ❧➝♥ ♥➳✉ dim R/p = dim R
✈ỵ✐ ♠å✐ p ssR

ỗ
r tt ú tổ tr ❜➔② ♠ët sè ❦➳t q✉↔ t❤❡♦ ❬✶✺❪✳



sỷ f : R S ỗ ✈➔♥❤✳ ❑❤✐ ✤â ♠é✐ S ✲♠ỉ✤✉♥ L ✤➲✉ ❝â ❝➜✉
tró❝ ❧➔ R✲♠æ✤✉♥✱ tr♦♥❣ ✤â ♣❤➨♣ ❝ë♥❣ ✤➣ ❝â s➤♥ tr♦♥❣ L t ổ ữợ

ừ tỷ r R ợ tỷ a L ữủ t f (r)a. ❈➜✉ tró❝
R✲♠ỉ✤✉♥ L ①→❝ ✤à♥❤ ♥❤÷ t❤➳ ✤÷đ❝ ồ trú ổ

ởt ỗ f : R S ữủ ồ ỗ ♣❤➥♥❣ ♥➳✉ S ①➨t ♥❤÷
R✲♠ỉ✤✉♥ ①→❝ ✤à♥❤ ❜ð✐ f ❧➔ R✲♠ỉ✤✉♥ ♣❤➥♥❣✱ tù❝ ❧➔ ✈ỵ✐ ♠é✐ ❞➣② ❦❤ỵ♣
0→L →L→L →0

❝→❝ R✲♠æ✤✉♥✱ ❞➣② ❝↔♠ s✐♥❤ 0 → L ⊗ S → L ⊗ S → L ⊗ S → 0 ❧➔ ợ
ởt ỗ f : R S ữủ ồ ỗ t S
t ữ Rổ ①→❝ ✤à♥❤ ❜ð✐ f ❧➔ R✲♠æ✤✉♥ ❤♦➔♥ t♦➔♥ ♣❤➥♥❣✱ tù❝ ❧➔ ✈ỵ✐
♠é✐ ❞➣②
0→L →L→L →0

❝→❝ R✲♠ỉ✤✉♥ ❧➔ ❦❤ỵ♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❞➣② ❝↔♠ s✐♥❤
0→L ⊗S →L⊗S →L ⊗S →0

❧➔ ❦❤ỵ♣.

▼➺♥❤ ✤➲ ✶✳✶✵✳✶✳ ❈→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣✿

✭✐✮ ◆➳✉ f : R S ỗ t ♣❤➥♥❣ t❤➻ →♥❤ ①↕ ❝↔♠ s✐♥❤
af : ❙♣❡❝S → ❙♣❡❝R ❝❤♦ ❜ð✐ af (p) = f −1 (p) := p ∩ R ✈ỵ✐ p ∈ ❙♣❡❝S ❧➔
t♦➔♥ →♥❤✳
✭✐✐✮ ◆➳✉ f : R S ỗ t ✈➔ L ❧➔ R✲♠æ✤✉♥ ❦❤→❝
✵ t❤➻ L ⊗R S ❧➔ S ✲♠æ✤✉♥ ❦❤→❝ ✵.
❈❤♦ f : R → S ❧➔ ỗ ỳ tr õ r f tọ
ỵ ợ ộ ♥❣✉②➯♥ tè q ⊂ p ❝õ❛ R


✷✹

✈ỵ✐ q = p ✈➔ ♠é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè P ❝õ❛ S s❛♦ ❝❤♦ f −1(P ) = p ✤➲✉ tỗ
t tố Q ừ S s Q ⊂ P ✈➔ f −1(Q) = q.

▼➺♥❤ ✤➲ ✶✳✶✵✳✷✳ ❈❤♦ (R, m) ✈➔ (S, n) ❧➔ ❝→❝ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ữỡ
f : R S ỗ ✤à❛ ♣❤÷ì♥❣ ✭tù❝ ❧➔ f (m) ⊆ n) t❤➻ f
t❤ä❛ ỵ

sỷ f : R S ởt ỗ ❣✐ú❛ ❝→❝
✈➔♥❤ ◆♦❡t❤❡r ✈➔ M ❧➔ ♠ët S ✲♠æ✤✉♥✳ ❑❤✐ õ

ssR(M ) = af (ssS (M )).

ỵ f

ởt ỗ ỳ
tr E ❧➔ ♠ët R✲♠æ✤✉♥ ✈➔ F ❧➔ ♠ët S ✲♠æ✤✉♥✳ ●✐↔ sû F ❧➔
R✲♠æ✤✉♥ ♣❤➥♥❣✳ ❑❤✐ ✤â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ✤➙② ❧➔ ✤ó♥❣✳
✭✐✮ ❱ỵ✐ ❜➜t ❦➻ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ❝õ❛ R✱
: R → S

af (❆ssS (F/pF )) = ❆ssA (F/pF ) =




{p}


∅

✭✐✐✮ ❆ssB (E ⊗A F ) =

♥➳✉ F/pF = 0.

F/pF = 0.

ssB (F/pF ).
pssR (E)

ú ỵ r ỗ tỹ R R t ♣❤➥♥❣✳ ❈❤ó♥❣ t❛
❝â t❤➸ sû ❞ư♥❣ ▼➺♥❤ ✤➲ ✶✳✶✵✳✸ ✈➔ ỵ ữủ t q s

❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣ ✈➔ M ❧➔ R✲♠æ✤✉♥
❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â
✭✐✮ ❆ssR(M ) =

p∈❆ssM

❆ssR(R/pR).

✭✐✐✮ ❆ss(M ) = {q ∩ R | q ∈ ❆ssR(M )}.