❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❯♠❛ ■♥tr♦❞✉çã♦ às ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s s♦❜r❡ ❛
➪❧❣❡❜r❛ ❞❡ ▼❛tr✐③❡s ❚r✐❛♥❣✉❧❛r❡s ❙✉♣❡r✐♦r❡s
▼❛t❡✉s ❊❞✉❛r❞♦ ❙❛❧♦♠ã♦
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❯♠❛ ■♥tr♦❞✉çã♦ às ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s s♦❜r❡ ❛
➪❧❣❡❜r❛ ❞❡ ▼❛tr✐③❡s ❚r✐❛♥❣✉❧❛r❡s ❙✉♣❡r✐♦r❡s
▼❛t❡✉s ❊❞✉❛r❞♦ ❙❛❧♦♠ã♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉✐♠❛s ❏♦sé ●♦♥ç❛❧✈❡s
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥✲ çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico
com os dados fornecidos pelo(a) autor(a)
S173i
Salomão, Mateus Eduardo
Uma introdução às identidades funcionais sobre a álgebra de matrizes triangulares superiores / Mateus Eduardo Salomão. -- São Carlos : UFSCar, 2016. 96 p.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2016.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣❡❧❛ ❢♦rç❛ q✉❡ ♠❡✉ ❞❡✉ ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❞♦ ♠❡str❛❞♦❀
➚ ♠✐♥❤❛ ♠ã❡ ■♦♥❡✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦ ♥❡st❡ t❡♠♣♦✱ ❡ ♣❡❧❛s ♦r❛çõ❡s q✉❡ ❛ ♠✐♠ ❞❡❞✐❝♦✉✳ ❚❛♠❜é♠ ❛❣r❛❞❡ç♦ ❛ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ q✉❡ t❛♠❜é♠ t♦r❝❡✉ ❡ r❡③♦✉ ♣♦r ♠✐♠❀
❆♦ ♣r♦❢❡ss♦r ❉✐♠❛s✱ ♣♦r t❡r ❛❝❡✐t❛❞♦ ♠❡ ♦r✐❡♥t❛r✱ ❡ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ q✉❡ t❡✈❡ ❝♦♠ ♦ ♠❡✉ tr❛❜❛❧❤♦✳ ■ss♦ s❡♠ ❝♦♥t❛r ♦s ❡♥s✐♥❛♠❡♥t♦s ❡ ❛✉①í❧✐♦s q✉❡ ♠❡ ❞❡✉ ❞✉r❛♥t❡ ♦ ♠❡str❛❞♦❀
❆♦s ♠❡✉s ❛♠✐❣♦s ❈❛r❧♦s✱ ❉❛✐❛♥❛ ❡ ❊✈❛♥❞r♦ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦✱ ❛✉①✐❧✐♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❢♦rç❛ q✉❡ ♠❡ ❞❡r❛♠ ❞✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦✳ ❆❣r❛❞❡ç♦ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ❊✈❛♥❞r♦✱ ♣❡❧❛ ❝♦♠♣❛♥❤✐❛✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s✱ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ❡st✉❞♦s ❡ ♠♦♠❡♥t♦s ❛❧❡❣r❡s ❞❡ ❞❡s❝♦♥tr❛çã♦❀
❆♦s ❞❡♠❛✐s ❛♠✐❣♦s✱ q✉❡ t♦r❝❡r❛♠ ♣♦r ♠✐♠✱ ♠❡ ❛♣♦✐❛r❛♠ ❡ ✐♥❝❡♥t✐✈❛r❛♠❀ ❆♦s ♠❡✉s ♣r♦❢❡ss♦r❡s✱ ♣❡❧❛ ❡①❝❡❧❡♥t❡ ❢♦r♠❛çã♦ ❞✐s♣❡♥s❛❞❛❀
❘❡s✉♠♦
❖ ❛ss✉♥t♦ tr❛t❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ ❞✐③ r❡s♣❡✐t♦ à ✐❞❡♥t✐❞❛❞❡s ❢✉♥❝✐♦♥❛✐s ✭❋■✮ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❛♥❡❧✳ ❙ã♦ ❢♦r♥❡❝✐❞♦s ❝♦♥❝❡✐t♦s✱ ❡①❡♠♣❧♦s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ♦s t❡♠❛s✿ s♦❧✉çã♦ st❛♥❞❛r❞ ❞❡ ✉♠❛ ❋■✱ ❣r❛✉ ❢♦rt❡ ❞❡ ✉♠ ❛♥❡❧✱ ❛♥é✐s ❢♦rt❡♠❡♥t❡ d✲❧✐✈r❡s ❡ ❋■✲❣r❛✉ ❞❡ ✉♠ ❛♥❡❧✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ sã♦ ❡st✉❞❛❞❛s s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡s♣❡❝í✜❝❛ ❋■ ♣❛r❛ ❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r❡s✱ ✐st♦ é✿ ❙❡❥❛♠ r ❡ n ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❝♦♠ r ≥ 2✱ Tr ❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r❡s r×r s♦❜r❡ ✉♠ ❝♦r♣♦ F ❡
f : (Tr)n −→ Tr ✉♠❛ ❢✉♥çã♦ ♠✉❧t✐❧✐♥❡❛r t❛❧ q✉❡
[f(A, A, . . . , A), A] = 0, ♣❛r❛ t♦❞♦ A∈ Tr.
❙❡n≤r ❡|F|> n+ 1✱ ❡♥tã♦ t❛✐s ❢✉♥çõ❡sf sã♦ ❞❡s❝r✐t❛s✳
❆❜str❛❝t
❚❤❡ s✉❜❥❡❝t tr❡❛t❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s ❢✉♥❝t✐♦♥❛❧ ✐❞❡♥t✐t✐❡s ✭❋■✮ ♦❢ ❛ ❡s♣❡❝✐✜❝ r✐♥❣✳ ❲❡ ♣r❡s❡♥t ❝♦♥❝❡♣ts✱ ❡①❛♠♣❧❡s ❛♥❞ s♦♠❡ r❡s✉❧ts ✐♥✈♦❧✈✐♥❣ t❤❡ t❤❡♠❡s✿ st❛♥❞❛r❞ s♦❧✉t✐♦♥ ♦❢ ❛ ❋■✱ str♦♥❣ ❞❡❣r❡❡ ♦❢ ❛ r✐♥❣✱ str♦♥❣❧② ❞✲❢r❡❡ r✐♥❣s ❛♥❞ ❋■✲❞❡❣r❡❡ ♦❢ ❛ r✐♥❣✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐t ✐s st✉❞✐❡❞ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❋■ ♦♥ ✉♣♣❡r tr✐❛♥❣✉❧❛r ♠❛tr✐❝❡s ❛❧❣❡❜r❛✱ t❤❛t ✐s✿ ▲❡t r ❛♥❞ n ❜❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ✇✐t❤ r ≥ 2✱ Tr ❜❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ✉♣♣❡r
tr✐❛♥❣✉❧❛r r×r ♠❛tr✐❝❡s ♦✈❡r ❛ ✜❡❧❞ F ❛♥❞ f : (Tr)n −→ Tr ❜❡ ❛ ♠✉❧t✐❧✐♥❡❛r ♠❛♣♣✐♥❣
s✉❝❤ t❤❛t
[f(A, A, . . . , A), A] = 0, ❢♦r ❛❧❧ A∈ Tr.
■❢n ≤r ❛♥❞ |F|> n+ 1 t❤❡♥ f ✐s ❞❡s❝r✐❜❡❞✳
❙✉♠ár✐♦
✶ Pr❡❧✐♠✐♥❛r❡s ✸
✶✳✶ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s ✶✼
✷✳✶ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ❉❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ■❞❡♥t✐❞❛❞❡ ❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸ ❖ ●r❛✉ ❋♦rt❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✹ ❆♥é✐s ❋♦rt❡♠❡♥t❡ d✲❧✐✈r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✺ ❆♥é✐s ❋♦rt❡♠❡♥t❡ (t;d)✲❧✐✈r❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✷✳✻ ❆ ❞❡s✐❣✉❛❧❞❛❞❡ s✲❞❡❣(A)≤ ❋■✲❞❡❣(A) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✸ ❋■ ♣❛r❛ ❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s ✼✸
✸✳✶ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s ❡♠Tr(F)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
■♥tr♦❞✉çã♦
❖ ❛ss✉♥t♦ ❛ s❡r tr❛t❛❞♦ ♥❡st❛ ❞✐ss❡rt❛çã♦ é ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s ✭❋■✮✳ ❙❡♠ ♠✉✐t♦ r✐❣♦r✱ ❞❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞♦ q✉❡ ✈❡♠ ❛ s❡r ✉♠❛ ❋■ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❛♥❡❧A✳ ❈♦♥s✐✲ ❞❡r❡ ✉♠ ♣♦❧✐♥ô♠✐♦f =f(x1, . . . , xm, y1, . . . , yn)❡♠ ✈❛r✐á✈❡✐s ♥ã♦ ❝♦♠✉t❛t✐✈❛sx1, . . . , xm,
y1, . . . , yn ❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠Z✳ ◆❛ t❡♦r✐❛ ❞❡ ❋■ ♣r♦❝✉r❛✲s❡ ♣♦r ❢✉♥çõ❡sFi :Am −→A✱
i= 1, . . . , n✱ t❛✐s q✉❡
f(r1, . . . , rm, F1(r1, . . . , rm), . . . , Fn(r1, . . . , rm)) = 0 ✭✶✮
♣❛r❛ t♦❞♦s r1, . . . , rm ∈ A✳ ❈❛s♦ t❛✐s ❢✉♥çõ❡s ❡①✐st❛♠✱ ❞✐③❡♠♦s q✉❡ ❡❧❛s ❢♦r♠❛♠ ✉♠❛
s♦❧✉çã♦ ❞❛ ❋■ ✭✶✮✳
❖ ✐♥í❝✐♦ ❞❛ ❋■✲t❡♦r✐❛ ❞❡✉✲s❡ ♣♦r ✈♦❧t❛ ❞❡ ✶✾✾✵ ❝♦♠ ❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ❞❡ ▼❛t❡❥ ❇r❡➨❛r✳ ❉❡♣♦✐s✱ t❛♠❜é♠ ❝♦♠ ♦s tr❛❜❛❧❤♦s ❞❡ ❑♦♥st❛♥t✐♥ ❇❡✐❞❛r ❡ ▼✐❦❤❛✐❧ ❈❤❡❜♦t❛r✱ ❛ t❡♦r✐❛ ♣❛ss♦✉ ❛ s❡r ❢✉♥❞❛♠❡♥t❛❞❛ ❡ ❡①t❡♥s✐✈❛♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞❛✳ ❯♠❛ ❞❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❋■✲t❡♦r✐❛ sã♦ ❛s s♦❧✉çõ❡s ❞❛s ❝♦♥❥❡❝t✉r❛s ❞❡ ❍❡rst❡✐♥ s♦❜r❡ ❤♦♠♦♠♦r✜s♠♦s ❞❡ ▲✐❡ ❡ ❞❡r✐✈❛çõ❡s ❞❡ ▲✐❡ ❡♠ ❛♥é✐s ❛ss♦❝✐❛t✐✈♦s✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❞♦s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❞❛ ❋■✲t❡♦r✐❛✱ s✉❣❡r✐♠♦s ♦ ❧✐✈r♦ ❬✹❪✳
❯♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ❋■ é ❛q✉❡❧❛ ❞❡❞✉③✐❞❛ ❛ ♣❛rt✐r ❞❡ ♣♦❧✐♥ô♠✐♦sf ❞♦ t✐♣♦
f =X
i
y1ixi+
X
j
xjy2j.
❙♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s ♥❛s s♦❧✉çõ❡s ❞❡s❡❥❛❞❛s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛♥❡❧ ❢♦rt❡♠❡♥t❡ d✲❧✐✈r❡ ❡ ❞❡❞✉③✐r r❡❧❛çõ❡s ❡♥tr❡ ♦ ❣r❛✉ ❢♦rt❡ ❡ ♦ ❋■✲❣r❛✉ ❞♦ ❛♥❡❧A✳
◆♦ ❛rt✐❣♦ ❬✶❪ ♦s ❛✉t♦r❡s ❡st✉❞❛♠ ✐❞❡♥t✐❞❛❞❡s ❢✉♥❝✐♦♥❛✐s s♦❜r❡ ❛ á❧❣❡❜r❛ ❞❡ ♠❛tr✐③❡s tr✐❛♥❣✉❧❛r❡s s✉♣❡r✐♦r❡s✿ ❙❡❥❛♠r❡n✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❝♦♠r≥2✱Tr ❛ á❧❣❡❜r❛ ❞❛s ♠❛tr✐③❡s
■♥tr♦❞✉çã♦
t❛❧ q✉❡
[f(A, A, . . . , A), A] = 0, ♣❛r❛ t♦❞♦A∈ Tr.
❙❡ n ≤ r ❡ |F| > n+ 1 ❡♥tã♦ é ♣r♦✈❛❞♦ q✉❡ ❡①✐st❡♠ λ0 ∈ F ❡ ❢✉♥çõ❡s ♠✉❧t✐❧✐♥❡❛r❡s
λi : (Tr)i −→F✱ ♣❛r❛ i= 1, . . . , n✱ t❛✐s q✉❡
f(A, A, . . . , A) =
n
X
i=0
λi(A, A, . . . , A)An−i, ♣❛r❛ t♦❞♦A ∈ Tr.
❈♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ t❛❧ r❡s✉❧t❛❞♦✱ ❛✐♥❞❛ ❡♠ ❬✶❪ é ♠♦str❛❞♦ ♦ s❡❣✉✐♥t❡✿ ❙❡❥❛ F ✉♠
❝♦r♣♦ ❝♦♠ ❝❤❛r(F) 6= 2 ❡ |F| >3✳ ❊♥tã♦ t♦❞❛ ❢✉♥çã♦ ❧✐♥❡❛r ❜✐❥❡t✐✈❛ θ : Tr −→ Tr✱ ♦♥❞❡
r≥3✱ s❛t✐s❢❛③❡♥❞♦
[θ(A2), θ(A)] = 0
♣❛r❛ t♦❞♦A ∈ Tr✱ é ❞❛ ❢♦r♠❛
θ(A) =λϕ(A) +µ(A)Idr,
♦♥❞❡ λ ∈ F✱ λ 6= 0✱ µ é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡♠ Tr ❡ ϕ é ✉♠ ❛✉t♦♠♦r✜s♠♦ ♦✉ ✉♠
❛♥t✐❛✉t♦♠♦r✜s♠♦ ❞❡Tr✳
❱❛❧❡ ❛ ♣❡♥❛ ♠❡♥❝✐♦♥❛r q✉❡ ♦s ❛✉t♦♠♦r✜s♠♦s ❡ ❛♥t✐❛✉t♦♠♦r✜s♠♦s ❞❡Trsã♦ ❞❡s❝r✐t♦s
❡♠ ❬✽❪ ❡ ❬✶✵✱ ❈♦r♦❧ár✐♦s ✻ ❡ ✼❪✳
❆ ❞✐ss❡rt❛çã♦ ❡stá ❞✐✈✐❞✐❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥♦ ❈❛♣ít✉❧♦ ✶ sã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❛♥é✐s ❡ ❞❛ t❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s✳ ❊ss❡s r❡s✉❧t❛❞♦s sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♦ ❚❡♦r❡♠❛ ❞❛ ❉❡♥s✐❞❛❞❡ ❞❡ ❏❛❝♦❜s♦♥✳ P❛r❛ ✉♠ ♠❡❧❤♦r ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞♦ ❛ss✉♥t♦ ❝✐t❛♠♦s ♦s ❧✐✈r♦s ❬✺✱ ✼✱ ✻✱ ✾❪✳ ◆♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♣♦✉❝♦ ❞❛ t❡♦r✐❛ ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s✳ ❖ ♠❛t❡r✐❛❧ ❢♦✐ ❡①tr❛í❞♦ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹✱ ❈❛♣ít✉❧♦s ✶ ❡ ✷❪✳ ◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❞♦ ❛rt✐❣♦ ❬✶❪✳
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ ❚❡♦✲ r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s ❡ ❞❛ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s✳ P❛r❛ ♠❛✐♦r❡s ✐♥❢♦r♠❛çõ❡s ❡ ✉♠ ❛♣r♦❢✉♥❞❛♠❡♥t♦ ❞♦ ❛ss✉♥t♦✱ s✉❣❡r✐♠♦s ❛s r❡❢❡rê♥❝✐❛s ❬✺❪✱ ❬✼❪✱ ❬✻❪ ❡ ❬✾❪✳
✶✳✶ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ♦s ❛♥é✐s ❝♦♥s✐❞❡r❛❞♦s s❡rã♦ ❛ss♦❝✐❛t✐✈♦s ❝♦♠ ♦✉ s❡♠ ✉♥✐❞❛❞❡ ❡ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♠✉t❛t✐✈♦s✳ ◗✉❛♥❞♦ ♦ ❛♥❡❧ t✐✈❡r ✉♥✐❞❛❞❡ s❡rá ❝❤❛♠❛❞♦ ❛♥❡❧ ✉♥✐tár✐♦✳
❆ s❡❣✉✐r✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ✭r❡❧❡♠❜r❛r✮ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s q✉❡ ❛♣❛r❡❝❡rã♦ ❝♦♠ ✉♠❛ ❝❡rt❛ ❢r❡q✉ê♥❝✐❛ ♥❛ ❞✐ss❡rt❛çã♦✳
P❛r❛ q✉❛❧q✉❡r ❛♥❡❧ A ❡ n ∈ N✱ ❞❡♥♦t❛r❡♠♦s ♣♦r Mn(A) ♦ ❛♥❡❧ ❞❛s ♠❛tr✐③❡s n×n
❝♦♠ ❡♥tr❛❞❛s ❡♠ A✳ ❈❛s♦ A s❡❥❛ ✉♠ ❛♥❡❧ ✉♥✐tár✐♦✱ ❡♥tã♦ Mn(A) ❝♦♥té♠ ❛s ❝❤❛♠❛❞❛s
♠❛tr✐③❡s ✉♥✐tár✐❛s✱ ✐st♦ é✱ ♠❛tr✐③❡s q✉❡ tê♠ ❡①❛t❛♠❡♥t❡ ✉♠❛ ❡♥tr❛❞❛ ✐❣✉❛❧ ❛1❡ ❛s ❞❡♠❛✐s
❡♥tr❛❞❛s ♥✉❧❛s✳ ❱❛♠♦s ❞❡♥♦t❛r ✉♠❛ ♠❛tr✐③ ✉♥✐tár✐❛ ❝♦♠ ❡♥tr❛❞❛ 1 ♥❛ ♣♦s✐çã♦ (i, j) ♣♦r
eij.
❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ A✳ ❖ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S é ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦♠❛s ❡ s✉❜tr❛çõ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ t✐♣♦
a1sa2, a1s, sa2 ❡ s,
♦♥❞❡a1, a2 ∈A ❡ s∈S✳ ❱❛♠♦s ❞❡♥♦t❛r ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣❡❧♦ s✉❜❝♦♥❥✉♥t♦S ♣♦r(S)✳
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❯♠ ✐❞❡❛❧ I ❞❡ A é ❞✐t♦ ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ s❡ I ❡stá ❝♦♥t✐❞♦ ♥♦ ❝❡♥tr♦ ❞❡ A✳
❘❡❧❡♠❜r❛♠♦s q✉❡ ♦ ❝❡♥tr♦ ❞❡ ✉♠ ❛♥❡❧ A✱ ❞❡♥♦t❛❞♦ ♣♦rZ(A)✱ é ♦ ❝♦♥❥✉♥t♦
Z(A) ={a∈A : ab=ba, ∀b∈A}.
❘❡❧❛❝✐♦♥❛❞♦ ❛♦ ❝♦♥❝❡✐t♦ ✏❝♦♠✉t❛r✑ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ❡❧❡♠❡♥t♦✱ ❝❤❛♠❛❞♦ ❝♦♠✉✲ t❛❞♦r✳
❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ a, b∈A✳ ❉❡✜♥✐♠♦s ♦ ❝♦♠✉t❛❞♦r ❞❡ a ❡ b ♣♦r
[a, b] =ab−ba.
❖❜s❡r✈❡ q✉❡ a∈ Z(A) s❡✱ ❡ s♦♠❡♥t❡ s❡✱[a, b] = 0 ♣❛r❛ t♦❞♦b ∈A✳
❆ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❞♦ ❝♦♠✉t❛❞♦r ❞❡a❡b t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ q✉❡ s❡rá ♠✉✐t♦ út✐❧ ♣❛r❛ ♥♦ss♦s ♣r♦♣ós✐t♦s✿
[ab, c] =a[b, c] + [a, c]b.
❉❡ ❢❛t♦✱ [ab, c] =abc−cab=abc−cab−acb+acb=a[b, c] + [a, c]b.
❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠ ❛♥❡❧ A é ❞✐t♦ ❛♥❡❧ ♣r✐♠♦ s❡ ♣❛r❛ q✉❛✐sq✉❡r ✐❞❡❛✐s I ❡ J ❞❡ A✱ IJ = 0 ✐♠♣❧✐❝❛I = 0 ♦✉J = 0✳
❯♠❛ ❝♦♥❞✐çã♦ ❡q✉✐✈❛❧❡♥t❡ à ❉❡✜♥✐çã♦ ✶✳✶✳✸ é q✉❡ ♣❛r❛ t♦❞♦s a, b ∈ A✱ aAb = 0
✐♠♣❧✐❝❛a = 0 ♦✉b = 0 ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❞❡ss❡ r❡s✉❧t❛❞♦ ✈❡r ❬✺✱ ▲❡♠❛ ✷✳✶✼❪✮✳
❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❯♠ ❛♥❡❧ A é ❞✐t♦ ❛♥❡❧ s❡♠✐♣r✐♠♦ s❡ ♣❛r❛ t♦❞♦ ✐❞❡❛❧ I ❞❡ A✱ I2 = 0
✐♠♣❧✐❝❛I = 0✳
❯♠❛ ❝♦♥❞✐çã♦ ❡q✉✐✈❛❧❡♥t❡ à ❉❡✜♥✐çã♦ ✶✳✶✳✹ é q✉❡ ♣❛r❛ t♦❞♦a∈A✱ aAa= 0 ✐♠♣❧✐❝❛
a= 0 ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ❞❡ss❡ r❡s✉❧t❛❞♦ ✈❡r ❬✺✱ ▲❡♠❛ ✷✳✷✶❪✮✳
◆❛ s❡q✉ê♥❝✐❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♠ó❞✉❧♦s✱ ♦ ❚❡♦r❡♠❛ ❞❛ ❉❡♥s✐❞❛❞❡ ❞❡ ❏❛❝♦❜s♦♥ ❡ ♦ ▲❡♠❛ ❞❡ ❙❝❤✉r✳ ❈♦♠♦ ♦s ♠ó❞✉❧♦s ❝♦♥s✐❞❡r❛❞♦s s❡rã♦ ❛♣❡♥❛s ♠ó❞✉❧♦s à ❡sq✉❡r❞❛✱ ✉s❛r❡♠♦s ❛♣❡♥❛s ♦ t❡r♠♦ ♠ó❞✉❧♦ ❡ ♦♠✐t✐r❡♠♦s ♦ ❛ ❡sq✉❡r❞❛✱ ❛ ♠❡♥♦s q✉❡ s❡❥❛ ♥❡❝❡ssár✐♦✳
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
✐✮ s✐♠♣❧❡s s❡ AM 6= 0 ❡ s❡ s❡✉s ú♥✐❝♦s s✉❜♠ó❞✉❧♦s sã♦ 0 ❡ M✳
✐✐✮ ✜❡❧ s❡ aM 6= 0 ♣❛r❛ t♦❞♦ 06=a∈A✳
❈♦♠ ❜❛s❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ❞❡✜♥✐♠♦s ❛♥❡❧ ♣r✐♠✐t✐✈♦ ❝♦♠♦ ❛❜❛✐①♦✿
❉❡✜♥✐çã♦ ✶✳✶✳✻✳ ❯♠ ❛♥❡❧ A é ❞✐t♦ ♣r✐♠✐t✐✈♦ ✭à ❡sq✉❡r❞❛✮ s❡ ❡①✐st❡ ✉♠ A✲♠ó❞✉❧♦ M s✐♠♣❧❡s ❡ ✜❡❧✳
P♦rt❛♥t♦✱ t❡♠♦s q✉❡ ✉♠ ❛♥❡❧Aé ♣r✐♠✐t✐✈♦ s❡ ❡①✐st❡ ✉♠ A✲♠ó❞✉❧♦M t❛❧ q✉❡M 6= 0
❡ ❝✉♠♣r❡ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✿ ✐✮ Am=M, ∀ 06=m ∈M. ✐✐✮ aM 6= 0, ∀06=a∈A.
❯♠ ❧❡♠❛ ❞❡ ❞❡♠♦♥str❛çã♦ s✐♠♣❧❡s✱ ♠❛s q✉❡ ❝✉♠♣r❡ ✉♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥❛ t❡♦r✐❛ é ♦ ❢❛♠♦s♦ ▲❡♠❛ ❞❡ ❙❝❤✉r✳ ❆♥t❡s ❞❡ ❡♥✉♥❝✐á✲❧♦✱ ❞❡♥♦t❛r❡♠♦s ♣♦rEndAM ♦ ❝♦♥❥✉♥t♦ ❞❡
t♦❞♦s ♦s ❡♥❞♦♠♦r✜s♠♦s ✭❞❡ ♠ó❞✉❧♦s✮ ❞❡ ✉♠A✲♠ó❞✉❧♦ M✳
▲❡♠❛ ✶✳✶✳✼ ✭▲❡♠❛ ❞❡ ❙❝❤✉r✮✳ ❙❡❥❛A ✉♠ ❛♥❡❧ ❡M ✉♠ A✲♠ó❞✉❧♦ s✐♠♣❧❡s✳ ❊♥tã♦ ♦ ❛♥❡❧ ❞❡ ❡♥❞♦♠♦r✜s♠♦s ∆ =EndAM é ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✺✱ ▲❡♠❛ ✸✳✺✵❪
❉❡✜♥✐çã♦ ✶✳✶✳✽✳ ❙❡❥❛ M ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦ ∆✱ ❡ s❡❥❛ A ✉♠ s✉❜❛♥❡❧ ❞❡End∆(M)✳ ❉✐③❡♠♦s q✉❡ Aé ✉♠ ❛♥❡❧ ❞❡♥s♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❞❡ M
s❡ ♣❛r❛ t♦❞♦n∈N✱ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡{u1, u2, . . . , un} ❞❡M✱ ❡
t♦❞♦ s✉❜❝♦♥❥✉♥t♦{v1, v2, . . . , vn} ❞❡ M✱ ❡①✐st❡ f ∈A t❛❧ q✉❡
f(u1) =v1, f(u2) = v2, . . . , f(un) =vn.
❈♦♠ ❜❛s❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝❛r❛❝t❡r✐③❛r ♦s ❛♥é✐s ♣r✐♠✐t✐✈♦s ❛ ♣❛rt✐r ❞♦ ♣ró①✐♠♦ r❡s✉❧t❛❞♦✳
❚❡♦r❡♠❛ ✶✳✶✳✾ ✭❚❡♦r❡♠❛ ❞❛ ❉❡♥s✐❞❛❞❡ ❞❡ ❏❛❝♦❜s♦♥✮✳ ❯♠ ❛♥❡❧ A é ♣r✐♠✐t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ é ✐s♦♠♦r❢♦ ❛ ✉♠ ❛♥❡❧ ❞❡♥s♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦✳
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ❛♣❡♥❛s ✉♠❛ ♣❛rt❡ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❛ q✉❡ s❡ r❡❢❡r❡ ❛ ❝♦♥str✉çã♦ ❞♦ ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t♦r✳
❙✉♣♦♥❤❛ q✉❡ A é ✉♠ ❛♥❡❧ ♣r✐♠✐t✐✈♦ ❡ s❡❥❛ M ✉♠ ❆✲♠ó❞✉❧♦ s✐♠♣❧❡s ❡ ✜❡❧✳ P❡❧♦ ▲❡♠❛ ✶✳✶✳✼ t❡♠♦s q✉❡ ∆ = EndAM é ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ M é ✉♠
A✲♠ó❞✉❧♦ ❡♥tã♦M é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❆ss✐♠✱ t❡♠♦s q✉❡M é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡
∆ ❝♦♠ ❛ ♦♣❡r❛çã♦
δm:=δ(m), ♦♥❞❡ δ∈∆ ❡m ∈M✳ ❆❣♦r❛✱ ❞❛❞♦δ ∈∆t❡♠♦s q✉❡
δ(am) = a(δm), ✭✶✳✶✮
♣❛r❛ t♦❞♦a ∈A ❡m ∈M✳
❉❡✜♥✐♠♦s✱ ♣❛r❛ ❝❛❞❛a∈A✱ ❛ ❢✉♥çã♦ a:M −→M ♣♦r a(m) =am,
♦♥❞❡ m∈M✳ P♦r ✭✶✳✶✮ s❡❣✉❡ q✉❡ a∈End∆M.
▲♦❣♦✱ ❛ ❛♣❧✐❝❛çã♦
A−→End∆M, a7→a,
é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳ ❆✐♥❞❛ ♠❛✐s✱ ❝♦♠♦M é ✜❡❧ ❛ ❛♣❧✐❝❛çã♦ ❛❝✐♠❛ é ✉♠ ❤♦♠♦✲ ♠♦r✜s♠♦ ❞❡ ❛♥é✐s ✐♥❥❡t♦r✳
P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ✏❞❡♥s✐❞❛❞❡✑ ❡ ♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s ✈❡r ❬✺✱ ❚❡♦r❡♠❛ ✺✳✶✻❪✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ ❞❡♥♦t❡ ♣♦r End(A) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡♥❞♦♠♦r✜s♠♦s ❞♦
❣r✉♣♦ ❛❞✐t✐✈♦ A✳ P❛r❛a, b∈A ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❜✐❧❛t❡r❛❧ aMb ∈End(A) ♣♦r
aMb(x) =axb.
❉❡✜♥✐çã♦ ✶✳✶✳✶✵✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❉❡✜♥✐♠♦s M(A) ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s
♦s ❡❧❡♠❡♥t♦s ❡♠ End(A) q✉❡ ♣♦❞❡♠ s❡r ❡s❝r✐t♦s ❝♦♠♦ ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ ♠✉❧t✐♣❧✐❝❛✲
çõ❡s ❜✐❧❛t❡r❛✐s aMb✳ ❚❡♠♦s q✉❡ M(A) é ✉♠ s✉❜❛♥❡❧ ❞❡ End(A)✱ ❝❤❛♠❛❞♦ ❞❡ ❛♥❡❧ ❞❡
♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ A✳
◆♦t❡ q✉❡ s❡f ∈ M(A)✱ ❡♥tã♦ ❡①✐st❡♠ a1, . . . , am, b1, . . . , bm ∈A t❛✐s q✉❡
f(x) =
m
X
k=1
akxbk, ∀x∈A.
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
❆❧é♠ ❞✐ss♦✱A é ✉♠ ♠ó❞✉❧♦ s♦❜r❡M(A) ❝♦♠ ♦♣❡r❛çã♦ ♣r♦❞✉t♦ ❞❡✜♥✐❞❛ ♣♦rf·x=f(x)
s❡f ∈ M(A) ❡x∈A✳
❆❣♦r❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ❛♥❡❧ s✐♠♣❧❡s✱ ❡ ♥❛ s❡q✉ê♥❝✐❛ ❞♦✐s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✶✳ ❯♠ ❛♥❡❧ A é ❞✐t♦ s✐♠♣❧❡s s❡ A2 6= 0 ❡ ♦s ú♥✐❝♦s ✐❞❡❛✐s ❞❡A sã♦0 ❡
A✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✶✷✳ ❙❡ A é ✉♠ ❛♥❡❧ s✐♠♣❧❡s ❡ ✉♥✐tár✐♦ ❡♥tã♦ A é ✉♠ ♠ó❞✉❧♦ s✐♠♣❧❡s s♦❜r❡ ♦ ❛♥❡❧M(A)✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ B ✉♠ s✉❜♠ó❞✉❧♦ ❞❡ A ❡ s✉♣♦♥❤❛ q✉❡ B 6= 0✳ ◗✉❡r❡♠♦s ♠♦str❛r
q✉❡B =A✳
❙❡❥❛ a∈ A✳ ❈♦♠♦ B 6= 0 ❡♥tã♦ ❡①✐st❡ b ∈B ♥ã♦ ♥✉❧♦✳ ❈♦♥s✐❞❡r❡ ♦ ✐❞❡❛❧ AbA6= 0✳
❊♥tã♦ ❝♦♠♦ A é s✐♠♣❧❡s t❡♠♦s q✉❡ AbA = A ❡ ❛ss✐♠✱ a = Pmk=1akbck ♣❛r❛ ❝❡rt♦s
a1, . . . , am, c1, . . . , cm ∈A✳
❉❡✜♥❛f ∈ M(A) ♣♦r f(x) =
m
X
k=1
akxck✳ ❚❡♠♦s q✉❡
f·b =f(b) =
m
X
k=1
akbck =a∈B.
▲♦❣♦✱ A=B✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✶✸✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ✉♥✐tár✐♦✳ ❖ ❛♥❡❧ EndM(A)A é ✐s♦♠♦r❢♦ ❛ Z(A)✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ f ∈ EndM(A)A ❡ s❡❥❛♠ a, b ∈ A✳ ❈♦♠♦ f é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡
♠ó❞✉❧♦s t❡♠♦s ♣❛r❛ t♦❞♦ x∈A q✉❡ ❛ ♣r❡♠✐ss❛ ❡ ❛ ✐♠♣❧✐❝❛çã♦ ❛❜❛✐①♦ ✈á❧✐❞❛s✿
f(aMbx) = aMbf(x)⇒f(axb) = af(x)b.
❋❛③❡♥❞♦x= 1 ❡ a= 1 ♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ t❡♠♦s
f(b) = f(1b) = f(1)b=f(1)Id(b).
❆❧é♠ ❞✐ss♦✱
f(1)b=f(1b) =f(b1) =bf(1). ▲♦❣♦✱ f(1) ∈ Z(A) ❡f =f(1)Id✳
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ ❛ ❢✉♥çã♦
ϕ:Z(A)−→EndM(A)A
❞❡✜♥✐❞❛ ♣♦r ϕ(r) =rId é ✉♠ ✐s♦♠♦r✜s♠♦✳
P❛r❛ ✜♥❛❧✐③❛r ❡st❛ s❡çã♦ ❞❡✜♥✐r❡♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❢✉♥çã♦ ❛❞✐t✐✈❛✱ ❢✉♥çã♦n✲ ❛❞✐t✐✈❛ ❡ ♦ tr❛ç♦ ❞❡ ✉♠❛ ❢✉♥çã♦✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦
Gn=G×G× · · · ×G
| {z }
n−vezes
,
♦♥❞❡ é G✉♠ ❣r✉♣♦✳
❉❡✜♥✐çã♦ ✶✳✶✳✶✹✳ ❙❡❥❛♠ G ❡H ❞♦✐s ❣r✉♣♦s ❛❞✐t✐✈♦s✳
✐✮ ❯♠❛ ❢✉♥çã♦ F : G −→ H é ❞✐t❛ ❛❞✐t✐✈❛ s❡ F(a+b) = F(a) +F(b), ♣❛r❛ t♦❞♦s a, b∈G✳
✐✐✮ ❯♠❛ ❢✉♥çã♦ F : Gn −→ H é ❞✐t❛ n✲❛❞✐t✐✈❛ s❡ ❢♦r ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛ ❡♠ ❝❛❞❛
❛r❣✉♠❡♥t♦✱ ✐st♦ é✱
F(g1, . . . , gi +g′i, . . . , gn) =F(g1, . . . , gi, . . . , gn) +F(g1, . . . , g′i, . . . , gn),
♣❛r❛ t♦❞♦sg1, . . . , gi, gi′, . . . , gn∈G ❡ t♦❞♦ i= 1, . . . , n✳
❆ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ tr❛ç♦ ❞❡ ✉♠❛ ❢✉♥çã♦ n✲❛❞✐t✐✈❛✳ ❉❡✜♥✐çã♦ ✶✳✶✳✶✺✳ ❙❡❥❛♠ G ❡ H ❞♦✐s ❣r✉♣♦s ❛❞✐t✐✈♦s ❡ F : Gn −→ H ✉♠❛ ❢✉♥çã♦
n✲❛❞✐t✐✈❛✳ ❆ ❢✉♥çã♦
G → H
x 7→ F(x, x, . . . , x)
é ❞✐t❛ s❡r ♦ tr❛ç♦ ❞❡ F✳
❆ ♣❛rt✐r ❞❡st❛ ❞❡✜♥✐çã♦ ✈❛♠♦s ❢❛❧❛r ❞♦ ♣r♦❝❡ss♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ ♦ tr❛ç♦ ❞❛ ❢✉♥çã♦F ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r é ③❡r♦✳ ◆❡st❡ ❝❛s♦ t❡♠♦s
X
π∈Sn
F(xπ(1), xπ(2), . . . , xπ(n)) = 0, ✭✶✳✷✮
✶✳✶✳ ❚❡♦r✐❛ ❞❡ ❆♥é✐s ❆ss♦❝✐❛t✐✈♦s
♣❛r❛ t♦❞♦sx1, . . . , xn ∈G✱ ♦♥❞❡Sn ❞❡♥♦t❛ ♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❞❡ ♦r❞❡♠n✳ ❱❛♠♦s ✈❡r✐✜❝❛r
❝♦♠ ❞❡t❛❧❤❡s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ✭✶✳✷✮ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ♦s ❝❛s♦s q✉❛♥❞♦ n = 2 ❡ n = 3✳
◗✉❛♥❞♦ n= 2✱F é ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❛❞✐t✐✈❛✱ ❡♥tã♦ t❡♠♦s
F(x1, x2) +F(x2, x1) =F(x1+x2, x1+x2)−F(x1, x1)−F(x2, x2) = 0.
◗✉❛♥❞♦ n = 3✱ t❡♠♦s q✉❡ F é ✉♠❛ ❛♣❧✐❝❛çã♦ ✸✲❛❞✐t✐✈❛✳ ❆ss✐♠✱ s❡ s✉❜st✐t✉í♠♦s x ♣♦r x1+x2 ❡♠ F(x, x, x) = 0 ♦❜t❡♠♦s✿
F(x1+x2, x1+x2, x1+x2) = F(x1, x1, x2) +F(x1, x2, x1) +F(x2, x1, x1)
+ F(x1, x2, x2) +F(x2, x1, x2) +F(x2, x2, x1)
= 0
♣❛r❛ t♦❞♦sx1, x2 ∈G✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ s✉❜st✐t✉✐r♠♦s x ♣♦r x1+x3 ❡♠ F(x, x, x) = 0✱
♦❜t❡♠♦s✿
F(x1+x3, x1+x3, x1+x3) = F(x1, x1, x3) +F(x1, x3, x1) +F(x3, x1, x1)
+ F(x1, x3, x3) +F(x3, x1, x3) +F(x3, x3, x1)
= 0
❡ t❛♠❜é♠ ✱ q✉❛♥❞♦ s✉❜st✐t✉í♠♦sx ♣♦rx2+x3 ❡♠ F(x, x, x) = 0✱ t❡♠♦s
F(x2+x3, x2+x3, x2+x3) = F(x2, x2, x3) +F(x2, x3, x2) +F(x3, x2, x2)
+ F(x2, x3, x3) +F(x3, x2, x3) +F(x3, x3, x2)
= 0.
❉❛s ❡q✉❛çõ❡s ❛❝✐♠❛ ❡ ❞❡ F(x1, x1, x1) = 0, F(x2, x2, x2) = 0 ❡ F(x3, x3, x3) = 0✱
s❡❣✉❡ q✉❡
F(x1+x2+x3, x1+x2 +x3, x1+x2+x3) =
X
π∈S3
F(xπ(1), xπ(2), xπ(3)) = 0,
♣❛r❛ t♦❞♦sx1, x2, x3 ∈G✳
P❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡ n >3✱ ♦❜t❡♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ ✭✶✳✷✮ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳
❉✐③❡♠♦s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ✭✶✳✷✮ é ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❧✐♥❡❛r✐③❛çã♦ ❞❡F(x, x . . . , x) = 0✱
♣❛r❛ t♦❞♦x∈G✳
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
✶✳✷ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
◆❡st❛ s❡çã♦ r❡❝♦r❞❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ❛ t❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s ✭❞♦ ✐♥❣❧ês P♦❧②♥♦♠✐❛❧ ■❞❡♥t✐t②✮ ❡ ✐♥tr♦❞✉③✐♠♦s ❛❧❣✉♠❛s ♥♦t❛çõ❡s q✉❡ s❡rã♦ ✉s❛❞❛s ♥♦ ❞❡❝♦rr❡r ❞❛ ❞✐ss❡rt❛çã♦✳
❉❡♥♦t❛r❡♠♦s ♣♦r F ✉♠ ❝♦r♣♦ q✉❛❧q✉❡r ❡ ♣♦r N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳
❆♦ ❧♦♥❣♦ ❞❡st❛ s❡çã♦✱ ❛s á❧❣❡❜r❛s ❝♦♥s✐❞❡r❛❞❛s s❡rã♦ ❛ss♦❝✐❛t✐✈❛s✱ ❝♦♠ ✉♥✐❞❛❞❡ ❡ s♦❜r❡ F✳
❙❡❥❛X ={x1, x2, . . .}✉♠ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❡♥✉♠❡rá✈❡❧ ❞❡ ✈❛r✐á✈❡✐s✳ ❉❡♥♦t❛♠♦s ♣♦r FhXi ❛ á❧❣❡❜r❛ ❛ss♦❝✐❛t✐✈❛ ❧✐✈r❡ ❝♦♠ ✉♥✐❞❛❞❡✱ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞❛ ♣♦r X✱ ✐st♦ é✱ FhXit❡♠ ✉♠❛ ❜❛s❡ ❢♦r♠❛❞❛ ♣♦r 1❡ ♣❡❧❛s ♣❛❧❛✈r❛s
xi1· · ·xin, xij ∈X, n ∈N,
❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♣♦r
(xi1· · ·xin)(xj1· · ·xjm) =xi1· · ·xinxj1· · ·xjm.
❖ ❡❧❡♠❡♥t♦s ❞❡ FhXi sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦❧✐♥ô♠✐♦s✳
❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ ❡f(x1, . . . , xn) = f ∈FhXi.❉✐③❡♠♦s q✉❡ f é ✉♠❛
✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ R s❡
f(r1, . . . , rn) = 0,
♣❛r❛ t♦❞♦s r1, . . . , rn ∈ R✳ ❉❡♥♦t❛♠♦s ♣♦r T(R) ♦ ❝♦♥❥✉♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s
❞❡R✳ ❙❡ T(R)6={0} ❞✐③❡♠♦s q✉❡ R é ✉♠❛ P■✲á❧❣❡❜r❛✳
❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ ❝♦♠✉t❛t✐✈❛✳ ❊♥tã♦ t❡♠♦s q✉❡ ♦ ❝♦♠✉t❛❞♦r
[x1, x2] =x1x2−x2x1
é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛R✳ ▲♦❣♦✱ R é ✉♠❛ P■✲á❧❣❡❜r❛✳
❊①❡♠♣❧♦ ✶✳✷✳✸✳ ❙❡❥❛ R ✉♠❛ á❧❣❡❜r❛ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❝♦♠ dim(R) < n✳ ❊♥tã♦ ♦ P♦❧✐♥ô♠✐♦ ❙t❛♥❞❛r❞ ❞❡ ❣r❛✉ n
Stn(x1, x2, . . . , xn) =
X
σ∈Sn
(−1)σxσ(1)xσ(2)· · ·xσ(n),
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
♦♥❞❡Sné ♦ ❣r✉♣♦ ❞❛s ♣❡r♠✉t❛çõ❡s ❞❡{1,2, . . . , n}❡(−1)σ é ♦ s✐♥❛❧ ❞❡σ✱ é ✉♠❛ ✐❞❡♥t✐❞❛❞❡
♣♦❧✐♥♦♠✐❛❧ ♣❛r❛R✳ P♦rt❛♥t♦✱R é ✉♠❛ P■✲á❧❣❡❜r❛✳ P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✻✱ ❊①❡♠♣❧♦ ✷✳✶✳✸❪✳
❙❡❣✉❡ ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r q✉❡ t♦❞♦ ♣♦❧✐♥ô♠✐♦ st❛♥❞❛r❞ ❞❡ ❣r❛✉≥n2+1é ✐❞❡♥t✐❞❛❞❡
♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ Mn(F)✱ ✈✐st♦ q✉❡ dim(Mn(F)) = n2. ❆❧é♠ ❞✐ss♦✱ ❡♠ ✶✾✺✵✱ ❆♠✐ts✉r ❡
▲❡✈✐t③❦✐ ❞❡♠♦♥str❛r❛♠ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✿
❚❡♦r❡♠❛ ✶✳✷✳✹ ✭❚❡♦r❡♠❛ ❞❡ ❆♠✐ts✉r✲▲❡✈✐t③❦✐✮✳ ❆ á❧❣❡❜r❛ Mn(F) s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡
st❛♥❞❛r❞ ❞❡ ❣r❛✉ 2n
St2n(x1, x2, . . . , x2n) =
X
σ∈S2n
(−1)σxσ(1)xσ(2)· · ·xσ(2n).
❆❧é♠ ❞✐ss♦✱ 2n é ♦ ❣r❛✉ ♠í♥✐♠♦ ❞❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ Mn(F)✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✻✱ Pá❣✐♥❛ ✽✵ ♦✉ ✽✷❪✳ ❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❯♠ ✐❞❡❛❧I ❞❡FhXi é ❝❤❛♠❛❞♦ ❞❡ ❚✲✐❞❡❛❧ s❡
ϕ(I)⊆I
♣❛r❛ t♦❞♦ ❡♥❞♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛sϕ :FhXi →FhXi✳
P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❡FhXi✱ t❡♠♦s q✉❡ ✉♠ ✐❞❡❛❧I é ✉♠ ❚✲✐❞❡❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱
f(g1, . . . , gn)∈I
♣❛r❛ t♦❞♦f(x1, . . . , xn)∈I ❡ g1, . . . , gn∈FhXi✳
❙❡ R é ✉♠❛ P■✲á❧❣❡❜r❛✱ ❡♥tã♦ T(R) é ✉♠ ❚✲✐❞❡❛❧✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣♦❞❡ s❡r ♠♦s✲
tr❛❞♦ q✉❡ s❡ I é ✉♠ ❚✲✐❞❡❛❧✱ ❡♥tã♦
T(FhXi/I) =I.
❊♥tã♦ ✉♠ ✐❞❡❛❧ é ✉♠ ❚✲✐❞❡❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ é ♦ ❝♦♥❥✉♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❞❡ ❛❧❣✉♠❛ P■✲á❧❣❡❜r❛✳
❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦S ❞❡FhXi✱ ❞✐③❡♠♦s q✉❡ ❛ ✐♥t❡rs❡çã♦ ❞♦s ❚✲✐❞❡❛✐s q✉❡ ❝♦♥tê♠ S é ♦ ❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S✳ ❊❧❡ é ♦ ♠❡♥♦r ❚✲✐❞❡❛❧ q✉❡ ❝♦♥té♠ S ❡ s❡rá ❞❡♥♦t❛❞♦ ♣♦r
hSiT✳ ❆❜❛✐①♦ ♦ ❞❡s❝r❡✈❡♠♦s✳
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
Pr♦♣♦s✐çã♦ ✶✳✷✳✻✳ ❙❡❥❛ S ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ FhXi✳ ❖ ❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r S é ❢♦r♠❛❞♦ ♣♦r t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ t✐♣♦
uf(g1, . . . , gn)v,
♦♥❞❡ u, g1, . . . , gn, v ∈FhXi ❡ f(x1, . . . , xn)∈S✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ✈❡r ❬✻✱ ❖❜s❡r✈❛çã♦ ✷✳✷✳✻❪✳
❊♠ ❣❡r❛❧✱ ❞❛❞♦ ✉♠ ❚✲✐❞❡❛❧✱ q✉❡r❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ✏❜♦♠✑ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✳ ❯♠ ♣♦❧✐♥ô♠✐♦f(x1, . . . , xm)∈FhXié ❤♦♠♦❣ê✲
♥❡♦ ❞❡ ❣r❛✉d❡♠xi✱ s❡ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♠♦♥ô♠✐♦s t❛✐s q✉❡ ❡♠ ❝❛❞❛ ♠♦♥ô♠✐♦
❞❡f✱ ❛ ✈❛r✐á✈❡❧xi ❛♣❛r❡❝❡d ✈❡③❡s✳ ❙❡ f(x1, . . . , xm)é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ di ❡♠ xi✱ ♣❛r❛
t♦❞♦ i = 1, . . . , m✱ ❞✐③❡♠♦s q✉❡ f(x1, . . . , xm) é ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ (d1, . . . , dm)✳
❯♠ ♣♦❧✐♥ô♠✐♦ ♠✉❧t✐✲❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ (1, . . . ,1) é ❝❤❛♠❛❞♦ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❣r❛✉ m✳
▲❡♠❛ ✶✳✷✳✼✳ ❙❡❥❛
f(x1, ..., xm) = n
X
i=0
fi(x1, ..., xm)∈FhXi,
♦♥❞❡ fi é ❛ ❝♦♠♣♦♥❡♥t❡ ❤♦♠♦❣ê♥❡❛ ❞❡ f ❞❡ ❣r❛✉ i ❡♠ x1✳ ❙❡ ♦ ❝♦r♣♦ F ❝♦♥té♠ ♠❛✐s q✉❡
n ❡❧❡♠❡♥t♦s✱ ❡♥tã♦
hf0, f1, ..., fniT =hfiT.
❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ F é ✐♥✜♥✐t♦✱ ❡♥tã♦ t♦❞♦ ❚✲✐❞❡❛❧ é ❣❡r❛❞♦ ♣♦r s❡✉s ❡❧❡♠❡♥t♦s ♠✉❧t✐✲
❤♦♠♦❣ê♥❡♦s✳
❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ hfiT ⊆ hf
0, f1, ..., fniT✳ ❆ss✐♠✱ ♣❛r❛ ♣r♦✈❛r ♦ ❧❡♠❛ é s✉✜❝✐❡♥t❡
♠♦str❛r q✉❡
f0, f1, ..., fn∈ hfiT.
❙❡❥❛♠ α0, α1, ..., αn ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ F✳ ❈♦♠♦ hfiT é ❚✲✐❞❡❛❧ t❡♠♦s q✉❡
f(αjx1, x2, ..., xm) = n
X
i=0
fi(αjx1, x2, ..., xm) = n
X
i=0
αjifi(x1, x2, ..., xm)∈ hfiT,
♣❛r❛ t♦❞♦ j = 0,1, ..., n✳
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
1 α0 α20 · · · αn0
1 α1 α21 · · · αn1
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
1 αn α2n · · · αnn
| {z }
A f0 f1 ✳✳✳ fn =
f(α0x1, x2, ..., xm)
f(α1x1, x2, ..., xm)
✳✳✳
f(αnx1, x2, ..., xm)
❆ ♠❛tr✐③A é ❛ ♠❛tr✐③ ❞❡ ❱❛♥❞❡r♠♦♥❞❡ q✉❡ t❡♠ ❞❡t❡r♠✐♥❛♥t❡
det(A) = Y
0≤i<j≤n
(αj−αi)6= 0.
▲♦❣♦✱ A é ✐♥✈❡rtí✈❡❧✳ ❆ss✐♠✱ f0 f1 ✳✳✳ fn =
b11 b12 · · · b1n+1
b21 b22 · · · b2n+1
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ bn+1 1 bn+1 2 · · · bn+1n+1
| {z }
A−1
f(α0x1, x2, ..., xm)
f(α1x1, x2, ..., xm)
✳✳✳
f(αnx1, x2, ..., xm)
❡ ♣♦rt❛♥t♦✱
f0, f1, ..., fn∈s♣❛♥{f(α0x1, x2, ..., xm), ..., f(αnx1, x2, ..., xm)} ⊆ hfiT.
❊♥t❡♥❞❡✲s❡ ♣♦r ✏s♣❛♥✑ ♦ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❣❡r❛❞♦ ♣❡❧♦ s✉❜❝♦♥❥✉♥t♦ ❡♠ q✉❡stã♦✳
❙❡ F é ✐♥✜♥✐t♦✱ ❡♥tã♦ ♣♦❞❡♠♦s ✉s❛r ♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛ s♦❜r❡ ❝❛❞❛ fi ♠❛s
❛❣♦r❛ ♥❛ ✈❛r✐á✈❡❧x2✳ ❆♣ós ❛❧❣✉♥s ♣❛ss♦s✱ t❡r❡♠♦s q✉❡ ♦ ❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦rf é ♦ ♠❡s♠♦
❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣❡❧♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❝♦♠♣♦♥❡♥t❡s ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛s ❞❡ f✱ ❝♦♥❝❧✉✐♥❞♦ ❛ss✐♠ ♦ r❡s✉❧t❛❞♦✳
◆♦t❡ q✉❡ ♣♦❞❡♠♦s ❞✐③❡r ❛❧❣♦ ♠❛✐s ❞♦ ❧❡♠❛ ❛❝✐♠❛✿ s❡ ♦ ❝♦r♣♦ F t❡♠ ♠❛✐s q✉❡ n ❡❧❡♠❡♥t♦s ❡ degxif < n ♣❛r❛ t♦❞♦i= 1, . . . , m✱ ❡♥tã♦ ♦ ❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦rf é ♦ ♠❡s♠♦ ❚✲✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r t♦❞❛s ❛s s✉❛s ❝♦♠♣♦♥❡♥t❡s ♠✉❧t✐✲❤♦♠♦❣ê♥❡❛s✳
▲❡♠❛ ✶✳✷✳✽✳ ❈♦♥s✐❞❡r❡ ✉♠ ♣♦❧✐♥ô♠✐♦
f(x1, . . . , xm) =
n
X
d1=0, ..., dm=0
α(d1, ..., dm)x
d1
1 · · ·xdmm ∈FhXi.
❙❡ f é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ F✱ ♦♥❞❡ F é ✉♠ ❝♦r♣♦ ❝♦♠ |F| ≥ n+ 1✱ ❡♥tã♦
f = 0 ✭♣♦❧✐♥ô♠✐♦ ♥✉❧♦✮✳
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
❉❡♠♦♥str❛çã♦✳ P❡❧♦ ▲❡♠❛ ✶✳✷✳✼ ❝❛❞❛ ♠♦♥ô♠✐♦α(d1, ..., dm)x
d1
1 · · ·xdmm ❞❡f t❛♠❜é♠ é ✉♠❛
✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ F✳ ❙✉❜st✐t✉✐♥❞♦ t♦❞♦s ♦sxi✬s ♣♦r1t❡♠✲s❡ q✉❡α(d1, ..., dm) = 0
❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ f = 0.
Pr♦♣♦s✐çã♦ ✶✳✷✳✾✳ ❙❡ ✉♠❛ á❧❣❡❜r❛ A s❛t✐s❢❛③ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ f✱ ❡♥tã♦ A t❛♠❜é♠ s❛t✐s❢❛③ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❣r❛✉ ≤ ❣r❛✉(f)✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ f =f(x1, . . . , xn) ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❞❡ A✳ ❉❡♥♦t❡ ♣♦r di ♦
❣r❛✉ ❞❡f ❡♠ xi✳ ❆ ♣r♦✈❛ s❡rá ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ ❡♠ d= max{d1, . . . , dn}>0.
❙❡ d = 1✱ ❡♥tã♦ ❝❛❞❛ xi ❛♣❛r❡❝❡ ❡♠ ❝❛❞❛ ♠♦♥ô♠✐♦ ❞❡ f ♥♦ ♠á①✐♠♦ ✉♠❛ ✈❡③✳
❖❜s❡r✈❡ q✉❡xi ♣♦❞❡ ♥ã♦ ❛♣❛r❡❝❡r ❡♠ ❝❛❞❛ ♠♦♥ô♠✐♦ ❞❡f✳ ❊♥tã♦f ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡
♠✉❧t✐❧✐♥❡❛r✳ ❙✉♣♦♥❤❛✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ q✉❡ λx1· · ·xm✱ ♦♥❞❡ λ é ✉♠ ❡s❝❛❧❛r
♥ã♦✲♥✉❧♦✱ é ✉♠ ♠♦♥ô♠✐♦ ❞❡ f ❞❡ ❣r❛✉ ♠✐♥✐♠❛❧✳ ❊♥tã♦ f(x1, . . . , xm,0, . . . ,0) é ✉♠❛
✐❞❡♥t✐❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ♥ã♦✲♥✉❧❛ ❞❡ ❣r❛✉ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛♦ ❣r❛✉ ❞❡f✳
❙✉♣♦♥❤❛ q✉❡ d > 1✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ s✉♣♦♥❤❛ q✉❡ ❡①✐st❡ k ≤ n t❛❧ q✉❡ dk = · · · = dn = d ❡ di < d ♣❛r❛ i < k✳ ❉❡✜♥❛ ✉♠ ♥♦✈♦ ♣♦❧✐♥ô♠✐♦ q✉❡ ❡♥✈♦❧✈❡ ✉♠❛
✈❛r✐á✈❡❧ ❛❞✐❝✐♦♥❛❧ g =g(x1, . . . , xn, xn+1) ♣♦r
g :=f(x1, . . . , xn−1, xn+xn+1)−f(x1, . . . , xn−1, xn)−f(x1, . . . , xn−1, xn+1).
◆♦t❡ q✉❡ g é t❛♠❜é♠ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ A✳ ❱❛♠♦s ❡s❝r❡✈❡r
f =X
i
λiwi,
♦♥❞❡ ♦swi✬s sã♦ ♠♦♥ô♠✐♦s ❞❡f ✭❞✐st✐♥t♦s ❡♥tr❡ s✐✮ ❡ ♦sλi✬s sã♦ ❡s❝❛❧❛r❡s ♥ã♦✲♥✉❧♦s✳ ❊♥tã♦
g ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦
g =X
i
λigi,
♦♥❞❡ gi é ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ wi ❞❛ ♠❡s♠❛ ❢♦r♠❛ ❝♦♠♦ g ❢♦✐ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❡ f✳ ❙❡
xn ♥ã♦ ❛♣❛r❡❝❡ ❡♠ wi✱ ❡♥tã♦ gi = −wi✳ ❙❡ xn ❛♣❛r❡❝❡ ❛♣❡♥❛s ✉♠❛ ✈❡③ ❡♠ wi✱ ❡♥tã♦
gi = 0✳ ❙❡ xn ❛♣❛r❡❝❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ✈❡③❡s ❡♠ wi✱ ❡♥tã♦ gi é ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s
♣♦ssí✈❡✐s ♠♦♥ô♠✐♦s ♦❜t✐❞♦s ♣❡❧❛ s✉❜st✐t✉✐çã♦ ❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠✱ ♠❛s ♥ã♦ t♦❞♦s✱ ❞♦sxn✬s
❡♠ wi ♣♦r xn+1✳ ❆ss✐♠✱ s❡ ❡♠ q✉❛❧q✉❡r ✉♠ ❞❡ss❡s ♠♦♥ô♠✐♦s s✉❜st✐t✉✐r♠♦s xn+1 ♣♦r xn✱
❡♥tã♦ t❡♠♦s ♥♦✈❛♠❡♥t❡ ♦ ♠♦♥ô♠✐♦ wi✳ P♦rt❛♥t♦✱ ♦s ♠♦♥ô♠✐♦s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ gi sã♦
✶✳✷✳ ❚❡♦r✐❛ ❞❡ P■✲á❧❣❡❜r❛s
❞✐❢❡r❡♥t❡s ❞♦s q✉❡ ❛♣❛r❡❝❡♠ ❡♠ gi′✱ s❡ i′ =6 i✳ ❈♦♠♦ d > 1✱ ❡①✐st❡♠ í♥❞✐❝❡s i t❛✐s q✉❡ xn
❛♣❛r❡❝❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ✈❡③❡s ❡♠ wi✳ ■ss♦ ♠♦str❛ q✉❡ g 6= 0✳
❆s s❡❣✉✐♥t❡s ❝♦♥❝❧✉sõ❡s ♣♦❞❡♠ s❡r ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✿ ✐✮ g é ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♥ã♦✲♥✉❧❛ ❞❡ A✳
✐✐✮ ❖ ❣r❛✉ ❞❡ g é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛♦ ❣r❛✉ ❞❡ f✳
✐✐✐✮ P❛r❛ j = 1, . . . , n−1✱ ♦ ❣r❛✉ ❞❡ g ❡♠ xj é ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ dj
✐✈✮ ❖ ❣r❛✉ ❞❡g ❡♠ xn ❡xn+1 é d−1✳
❘❡♣❡t✐♥❞♦ ❡ss❡ ♣r♦❝❡ss♦✱ ♣r✐♠❡✐r♦ ❝♦♠ g ♥♦ ❧✉❣❛r ❞❡ f ❡ xn−1 ♥♦ ❧✉❣❛r ❞❡ xn ❡✱
❡♠ s❡❣✉✐❞❛✱ ❝♦♠ ❛s ❞❡♠❛✐s ✈❛r✐á✈❡✐s ❛té xk✱ ❝❤❡❣❛r❡♠♦s ❡♠ ✉♠❛ s✐t✉❛çã♦ ♦♥❞❡ ✉♠❛
✐❞❡♥t✐❞❛❞❡ ♥ã♦✲♥✉❧❛ t❡♠ ❣r❛✉ ♥♦ ♠á①✐♠♦d−1❡♠ ❝❛❞❛ ✈❛r✐á✈❡❧ ❡ ✉s❛♠♦s ❛ ❤✐♣ót❡s❡ ❞❡
✐♥❞✉çã♦✳
❖ ♣r♦❝❡ss♦ ❞❡ ❝♦♥str✉çã♦ ❞♦ ♣♦❧✐♥ô♠✐♦ ♠✉❧t✐❧✐♥❡❛r ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ❞❡f✳
❈❛♣ít✉❧♦ ✷
■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❡ ✈ár✐♦s ❡①❡♠♣❧♦s ❞❡ ✐❞❡♥t✐❞❛❞❡s ❢✉♥❝✐♦♥❛✐s✳ ❖ ❛ss✉♥t♦ ❛ s❡r ❛♣r❡s❡♥t❛❞♦ ❢♦✐ ❡①tr❛í❞♦ ❞❛ r❡❢❡rê♥❝✐❛ ❬✹❪✳
✷✳✶ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ❛ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞♦ q✉❡ ✈❡♠ ❛ s❡r ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧✱ ❢❛r❡♠♦s ✉♠❛ s❡çã♦ ❝♦♠ ✈ár✐♦s ❡①❡♠♣❧♦s ❛ ✜♠ ❞❡ q✉❡ ♦ ❧❡✐t♦r s❡ ❢❛♠✐❧✐❛r✐③❡ ❝♦♠ ❛ ♥♦t❛çã♦ ❡ ❝♦♠ ♦ t❡♠❛✳
❉❛❞♦ ✉♠ ❛♥❡❧ A✱ ❝♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿
Pr♦❜❧❡♠❛✿ ◗✉❛✐s sã♦ ❛s ❢✉♥çõ❡sE, F :A−→A t❛✐s q✉❡
E(x)y+F(y)x= 0, ∀ x, y ∈A ? ✭✷✳✶✮
P♦r ✉♠ ❛❜✉s♦ ❞❡ ❧✐♥❣✉❛❣❡♠✱ ♣❛r❛ ♦ ♠♦♠❡♥t♦✱ ❝❤❛♠❛r❡♠♦s ❛ ❡①♣r❡ssã♦ ✭✷✳✶✮ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ✭♦✉ ❞❡ ♠❛♥❡✐r❛ ❛❜r❡✈✐❛❞❛ ❋■✱ ❞♦ ✐♥❣❧ês ❢✉♥❝t✐♦♥❛❧ ✐❞❡♥t✐t✐❡s✮✳ ❖❜s❡r✈❡ q✉❡ ❛s ❢✉♥çõ❡sE ❡F ❢❛③❡♠ ♦ ♣❛♣❡❧ ❞❡ ✐♥❝ó❣♥✐t❛s✳ ❆ t❡♦r✐❛ ❞❡ ❋■ ❡st✉❞❛ ❛s ❢✉♥çõ❡s q✉❡ s❛t✐s❢❛③❡♠ ❝❡rt❛s ✐❞❡♥t✐❞❛❞❡s✱ ❝♦♠♦ ❡♠ ✭✷✳✶✮✳
❱❛♠♦s ❞❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s♦❧✉çõ❡s✱ ✐st♦ é✱ ❢✉♥çõ❡s E ❡ F q✉❡ s❛t✐s❢❛③❡♠ ❛ ❋■ ✭✷✳✶✮✳
❊①❡♠♣❧♦ ✷✳✶✳✶✳ ❯♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s ❞❡ s♦❧✉çã♦ ♣❛r❛ ✭✷✳✶✮ éE =F = 0✳ ❈❤❛♠❛r❡♠♦s
❡ss❛ s♦❧✉çã♦ ❞❡ s♦❧✉çã♦ st❛♥❞❛r❞✳ ❖ ❝♦♥❝❡✐t♦ ❣❡r❛❧ ❞❡ s♦❧✉çã♦ st❛♥❞❛r❞ ❞❡ ✉♠❛ ❋■ s❡rá ❢♦r♥❡❝✐❞♦ ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❊①❡♠♣❧♦ ✷✳✶✳✷✳ ❙❡ A é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ E = Id ❡ F = −Id✱ ♦♥❞❡ Id é ❛ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ A✱ sã♦ s♦❧✉çõ❡s ❞❛ ❋■ ✭✷✳✶✮✳
❉❡ ❢❛t♦✱
E(x)y+F(y)x=xy−yx= 0, ∀ x, y ∈A.
❆♦ ❧♦♥❣♦ ❞❡st❛ s❡çã♦✱ ❞❛❞♦ ✉♠ ❛♥❡❧ A q✉❛❧q✉❡r✱ ❞❡♥♦t❛r❡♠♦s ♣♦r Z ♦ s❡✉ ❝❡♥tr♦
Z(A)✳
❊①❡♠♣❧♦ ✷✳✶✳✸✳ ❙❡❥❛ A✉♠ ❛♥❡❧ ❝♦♠ ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ ♥ã♦ ♥✉❧♦ I✳ ❉❛❞♦ q✉❛❧q✉❡r c∈I✱ ❛s ❢✉♥çõ❡s E(x) =−F(x) =cx sã♦ s♦❧✉çõ❡s ❞❛ ❋■ ✭✷✳✶✮✳
❈♦♠ ❡❢❡✐t♦✱
E(x)y+F(y)x=cxy−cyx=ycx−ycx= 0, ∀ x, y ∈A.
◆♦t❡ q✉❡ ❛ ♣❡♥ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ s❡❣✉❡ ❞❡ cx∈ I ⊂ Z ✭❡♥tã♦ cxy =
ycx✮ ❡ ❞❡ c∈I ⊂ Z ✭❡♥tã♦ cy =yc✮✳
❆ ♣❛rt✐r ❞❛ ✐❞❡♥t✐❞❛❞❡ ✭✷✳✶✮ ♣♦❞❡✲s❡ ❡♥❝♦♥tr❛r ✉♠❛ ♦✉tr❛ ✐❞❡♥t✐❞❛❞❡ q✉❡ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ ✉♠❛ ❞❛s ❞✉❛s ❢✉♥çõ❡s✳ P❛r❛ ✐ss♦✱ s✉♣♦♥❤❛ q✉❡ A é ✉♠ ❛♥❡❧ q✉❛❧q✉❡r✳ ❚❡♠♦s ♣❛r❛ t♦❞♦sx, y, z, w ∈A ❛s s❡❣✉✐♥t❡s ✐♠♣❧✐❝❛çõ❡s✿
E(x)y+F(y)x= 0 ⇒ E(x)yz =−F(yz)x
⇒ (E(x)yz)w=−F(yz)xw=E(xw)yz =−F(y)xwz =E(x)ywz
⇒ E(x)yzw−E(x)ywz = 0
⇒ E(x)y[z, w] = 0.
P♦rt❛♥t♦✱
E(A)A[A, A] = 0. ✭✷✳✷✮ ❯s❛r❡♠♦s ❡ss❛ ✐♥❢♦r♠❛çã♦ ♥♦s ❡①❡♠♣❧♦s ❛❜❛✐①♦✳
❊①❡♠♣❧♦ ✷✳✶✳✹✳ ❙❡ A é ✉♠ ❛♥❡❧ ♣r✐♠♦ ❡ ♥ã♦ ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞❛ ❋■ ✭✷✳✶✮ é ❛ st❛♥❞❛r❞✱ ✐st♦ é✱ E =F = 0.
❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡ E 6= 0. ❊♥tã♦ ❡①✐st❡ x ∈ A t❛❧ q✉❡ E(x) 6= 0. ❉❛❞♦s y, z ∈A t❡♠♦s ❞❡ ✭✷✳✷✮ ❛ ✐❣✉❛❧❞❛❞❡
E(x)A[y, z] = 0.
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
▼❛s ❝♦♠♦ A é ♣r✐♠♦ s❡❣✉❡ q✉❡ E(x) = 0 ♦✉ [y, z] = 0✳ ▲♦❣♦ [y, z] = 0, ∀ y, z ∈ A✳
❈♦♥tr❛❞✐çã♦✱ ♣♦✐sA ♥ã♦ é ❝♦♠✉t❛t✐✈♦✳ P♦rt❛♥t♦✱ E = 0.
❋❛③❡♥❞♦ ✉♠ ♣r♦❝❡ss♦ ❛♥á❧♦❣♦ t❡♠♦s q✉❡ F = 0. ❇❛st❛ ♥♦t❛r q✉❡ ✭✷✳✷✮ t❛♠❜é♠ é ✈á❧✐❞♦ tr♦❝❛♥❞♦E ♣♦r F✳
❆ ♣❛rt✐r ❞♦s ❡①❡♠♣❧♦s ✷✳✶✳✷ ❡ ✷✳✶✳✹✱ t❡♠♦s q✉❡ s❡ ✉♠ ❛♥❡❧ ♣r✐♠♦ é ❝♦♠✉t❛t✐✈♦ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ♥ã♦ st❛♥❞❛r❞ ❞❛ ❋■ ✭✷✳✶✮✱ ❡ s❡ ✉♠ ❛♥❡❧ ♣r✐♠♦ é ♥ã♦ ❝♦♠✉t❛t✐✈♦ ❡♥tã♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞❛ ❋■ ✭✷✳✶✮ é ❛ st❛♥❞❛r❞✳
❊①❡♠♣❧♦ ✷✳✶✳✺✳ ❙❡❥❛♠ A ✉♠ ❛♥❡❧ s❡♠✐♣r✐♠♦ ❡ E, F s♦❧✉çõ❡s ❞❡ ✭✷✳✶✮✱ ♦♥❞❡ E 6= 0✳ ❙❡
I = (E(A))✱ ❡♥tã♦I é ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ ❞❡A✳ P❛r❛ ♣r♦✈❛r ✐ss♦✱ ♣r✐♠❡✐r♦ ♠♦str❛r❡♠♦s q✉❡
[I, A]A[I, A] = 0.
❖❜s❡r✈❡ q✉❡ [I, A] =IA−AI ⊆I ♣♦✐sI é ✐❞❡❛❧ ❞❡ A✳ ❆ss✐♠✱
[I, A]A[I, A]⊆IA[A, A].
P♦rt❛♥t♦✱ ❛ ♣r✐♠❡✐r❛ ❛✜r♠❛çã♦ s❡ r❡s✉♠❡ ❛ ♠♦str❛r q✉❡
IA[A, A] = 0.
❙❛❜❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❡I sã♦ s♦♠❛s ❡ s✉❜tr❛çõ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛asb, as, sb ❡s✱ ♦♥❞❡ a, b∈ A ❡ s∈E(A)✳ ▼♦str❛r❡♠♦s ❛♣❡♥❛s q✉❡ asbr[p, q] = 0✱ ♦♥❞❡ a, b, r, p, q ∈ A ❡ s ∈ E(A)✳ ❈♦♠♦ ♣♦r ✭✷✳✷✮ t❡♠♦s sbr[p, q] = 0✱ s❡❣✉❡ ♦ ❞❡s❡❥❛❞♦✳ Pr♦❝❡❞❡♥❞♦
❛♥❛❧♦❣❛♠❡♥t❡ ♣❛r❛ ♦s ❞❡♠❛✐s ❝❛s♦s✱ t❡♠♦s q✉❡IA[A, A] = 0.
❆❣♦r❛✱ ♣❛r❛ ♣r♦✈❛r q✉❡ I é ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ ❞❡ A✱ ❜❛st❛ ♠♦str❛r q✉❡ [I, A] = 0✳
❉❛❞♦ x ∈ [I, A]✱ ❞❡ [I, A]A[I, A] = 0 t❡♠♦s xAx = 0✳ ❈♦♠♦ A é s❡♠✐♣r✐♠♦✱ s❡❣✉❡ q✉❡ x= 0✳
❋❛③❡♥❞♦ ✉♠ ♣r♦❝❡ss♦ ❛♥á❧♦❣♦ ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡E, F sã♦ s♦❧✉çõ❡s ❞❡ ✭✷✳✶✮✱ ♦♥❞❡ F 6= 0✱ t❡♠♦s q✉❡ J = (F(A))é t❛♠❜é♠ ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ ❞❡ A✳
❉♦s ❡①❡♠♣❧♦s ✷✳✶✳✸ ❡ ✷✳✶✳✺ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ✉♠ ❛♥❡❧ s❡♠✐♣r✐♠♦ ❝♦♥té♠ ✉♠ ✐❞❡❛❧ ❝❡♥tr❛❧ ♥ã♦ ♥✉❧♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ♥ã♦ st❛♥❞❛r❞ ❞❡ ✭✷✳✶✮ ♣❛r❛ ❡ss❡ ❛♥❡❧✳
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❊①❡♠♣❧♦ ✷✳✶✳✻✳ ❙❡❥❛ A = Mn(C)✱ n ≥ 2✱ ♦♥❞❡ C é ✉♠ ❛♥❡❧ ✉♥✐tár✐♦ ❝♦♠✉t❛t✐✈♦✳ ❆
ú♥✐❝❛ s♦❧✉çã♦ ❞❛ ❋■ ✭✷✳✶✮ é ❛ st❛♥❞❛r❞✱ ✐st♦ é✱ E =F = 0✳
❉❡ ❢❛t♦✱ s❡❥❛♠i6=j✱ ♦♥❞❡1≤i, j ≤n✳ ❉❡ ✭✷✳✷✮ t❡♠♦s ♣❛r❛ t♦❞♦x∈A♦ s❡❣✉✐♥t❡✿
E(x)eij[eji, eii] =E(x)eii= 0.
▲♦❣♦✱
E(x) =E(x)Idn =E(x)e11+E(x)e22+...+E(x)enn = 0,
❝♦♥❝❧✉✐♥❞♦ q✉❡E = 0✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❝♦♥❝❧✉í♠♦s q✉❡ F = 0✳
❉❛❞♦ ✉♠ ❛♥❡❧A✱ ❝♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿ Pr♦❜❧❡♠❛✿ ◗✉❛✐s sã♦ ❛s ❢✉♥çõ❡sE, F :A −→A t❛✐s q✉❡
( E(x)y+F(y)x) ) ∈ Z, ∀ x, y ∈A ? ✭✷✳✸✮
P♦r ✉♠ ❛❜✉s♦ ❞❡ ❧✐♥❣✉❛❣❡♠✱ ❝❤❛♠❛♠♦s ❛ ❡①♣r❡ssã♦ ✭✷✳✸✮ t❛♠❜é♠ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧✳ ❖❜s❡r✈❡ q✉❡ ❡st❛ é ❡q✉✐✈❛❧❡♥t❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧
[E(x)y+F(y)x, z] = 0, ∀ x, y, z ∈A.
❯♠❛ s♦❧✉çã♦ ❞❛ ❋■ ✭✷✳✸✮ é ❞❛❞❛ ♣♦r E = F = 0✱ t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ s♦❧✉çã♦
st❛♥❞❛r❞ ❞❡ t❛❧ ❋■✳
❆♥t❡s ❞❡ ♣r♦ss❡❣✉✐r✱ ❢❛r❡♠♦s ✉♠❛ ♣❛✉s❛ ❡ ❢❛❧❛r❡♠♦s ✉♠ ♣♦✉❝♦ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝✲ t❡ríst✐❝♦ ❞❡ ✉♠❛ ♠❛tr✐③✳ ■ss♦ s❡rá ♥❡❝❡ssár✐♦ ♣❛r❛ ❡st✉❞❛r ❛ ✐❞❡♥t✐❞❛❞❡ ❢✉♥❝✐♦♥❛❧ ✭✷✳✸✮ ❡♠ ♠❛tr✐③❡s ❝♦♠ ❡♥tr❛❞❛s ♥✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❡ ♣❛r❛ ♦✉tr❛s s✐t✉❛çõ❡s✳ ❙❡❥❛C ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❡ ❝♦♥s✐❞❡r❡ B ∈Mn(C)✳ ❖ ♣♦❧✐♥ô♠✐♦ ❡♠ C[λ] ❞❡✜♥✐❞♦ ♣♦r
pB(λ) = det(λIdn−B)
é ❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡B✳ ❊♠❜♦r❛ ♦ ❚❡♦r❡♠❛ ❞❡ ❈❛②❧❡②✲❍❛♠✐❧t♦♥ s❡❥❛ ❡♥✉♥❝✐❛❞♦ ♣❛r❛ ♠❛tr✐③❡s ❝♦♠ ❡♥tr❛❞❛s ♥✉♠ ❝♦r♣♦✱ ❡❧❡ ♣♦❞❡ s❡r ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛B✳
❚❡♦r❡♠❛ ✷✳✶✳✼ ✭❈❛②❧❡②✲❍❛♠✐❧t♦♥✮✳ ❙❡❥❛C ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❡ ❝♦♥s✐❞❡r❡B ∈Mn(C)✳
❙❡ pB(λ) é ♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ B✱ ❡♥tã♦ pB(B) = 0✳
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❉❡♠♦♥str❛çã♦✳ ❙❡Cé ✉♠ ❞♦♠í♥✐♦ ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ♣♦❞❡♠♦s ♠❡r❣✉❧❤❛rC♥♦ s❡✉ ❝♦r♣♦ ❞❡ ❢r❛çõ❡s K✳ ❈♦♠♦ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ ♣❛r❛ Mn(K)✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦ ♣r♦✈❛❞♦ ♥❡st❡
❝❛s♦✳ ❈♦♥s✐❞❡r❡ ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ❝♦♠✉t❛t✐✈♦
Z[x11, x12, . . . , xnn]
♥❛s ✈❛r✐á✈❡✐sx11, x12, . . . , xnn✳ ❈♦♠♦ ❡st❡ ❛♥❡❧ é ✉♠ ❞♦♠í♥✐♦ ❝♦♠✉t❛t✐✈♦✱ t❡♠♦s q✉❡
pB(B) = 0
♦♥❞❡ B = (xij)ij✳ ❙❡ C é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ q✉❛❧q✉❡r ❡ B = (bij)ij ∈ Mn(C)✱ ❡♥tã♦
pB(B)é ❡①❛t❛♠❡♥t❡pB(B)q✉❛♥❞♦ tr♦❝❛♠♦s ❛s ✈❛r✐á✈❡✐sxij ♣♦rbij✳ ▲♦❣♦✱pB(B) = 0✳
❊①❡♠♣❧♦ ✷✳✶✳✽✳ ❙❡❥❛ A = Mn(C)✱ ♦♥❞❡ C é ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ✉♥✐tár✐♦✳ ❊①✐st❡ ✉♠❛
s♦❧✉çã♦ ♥ã♦ st❛♥❞❛r❞ ❞❡ ✭✷✳✸✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱1≤n≤2. ❱❛♠♦s ✈❡r✐✜❝❛r t❛❧ ❛✜r♠❛çã♦✿
✐✮ P❛r❛n = 1t❡♠♦s A=C✳ ▲♦❣♦✱ E, F :C −→C ❞❡✜♥✐❞❛s ♣♦rE =Id ❡F =−Id ❢♦r♠❛♠ ✉♠❛ s♦❧✉çã♦ ♥ã♦ st❛♥❞❛r❞ ❞❛ ❋■✳
✐✐✮ P❛r❛ n = 2 t❡♠♦s A=M2(C)✳ ❙❡ x∈A✱ ♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡ x é
p(λ) = ❞❡t(λ(Id2)−x) =λ2−tr(x)λ+det(x).
P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈❛②❧❡②✲❍❛♠✐❧t♦♥ t❡♠✲s❡ p(x) = 0 ❡ ✈❛❧❡♠ ❛s ✐♠♣❧✐❝❛çõ❡s
p(x) = x2−tr(x)x+det(x)Id2 = 0 ⇒ x2−tr(x)x=−det(x)Id2 ∈ Z
⇒ [x2−tr(x)x, z] = 0, ∀ x, z ∈A.
❙✉❜st✐t✉✐♥❞♦x ♣♦r x+y ♥❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ t❡♠✲s❡
[(x+y)2−tr(x+y)(x+y), z] = 0.
❆❜r✐♥❞♦ ❛s ❝♦♥t❛s ❡ ❛❣r✉♣❛♥❞♦ ♦s ❢❛t♦r❡s✱ t❡♠♦s
[x2−tr(x)x, z] + [y2−tr(y)y, z] + [(x−tr(x)Id2)y+ (y−tr(y)Id2)x, z] = 0,
❈♦♠♦ ♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝♦♠✉t❛❞♦r❡s sã♦ ♥✉❧♦s✱ s❡❣✉❡ q✉❡
[(x−tr(x)Id2)y+ (y−tr(y)Id2)x, z] = 0.
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❆ss✐♠✱ E, F : A−→A ❞❛❞❛s ♣♦r E(x) =F(x) =x−tr(x)Id2 ❢♦r♠❛♠ ✉♠❛ s♦❧✉çã♦ ♥ã♦
st❛♥❞❛r❞ ❞❡ ✭✷✳✸✮ ♣❛r❛ A✳
✐✐✐✮ ❙✉♣♦♥❤❛ n≥3✳ ❉❡✜♥❛
π(x, y) =E(x)y+F(y)x
❡ s✉♣♦♥❤❛ q✉❡ π(x, y) ∈ Z✱ ♣❛r❛ t♦❞♦ x, y ∈ A✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ E = F = 0✳
Pr✐♠❡✐r♦ ❢❛r❡♠♦s ✐ss♦ ♣❛r❛ E✳ P❛r❛ t♦❞♦ x, y, t∈A t❡♠♦s
π(xt, y)−π(x, y)t =E(xt)y+F(y)xt−E(x)yt−F(y)xt=E(xt)y−E(x)yt.
❆❣♦r❛✱
[E(xt)y−E(x)yt, t] = [π(xt, y)−π(x, y)t, t] = [π(xt, y), t]−[π(x, y)t, t] = −[π(x, y)t, t]
= −[π(x, y), t]t
= 0.
❆ss✐♠✱ ❛❜r✐♥❞♦ ❛ ❡①♣r❡ssã♦ [E(xt)y−E(x)yt, t] = 0 t❡r❡♠♦s
−E(x)yt2+ (tE(x) +E(xt))yt−tE(xt)y = 0.
❉❛❞♦1≤i≤n✱ s✉❜st✐t✉❛t =e12+e23❡ y=ei1 ♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ ❡ ❞❡♣♦✐s ♠✉❧t✐♣❧✐q✉❡
❛ ❞✐r❡✐t❛ ♣♦r e3i✳ ❖ r❡s✉❧t❛❞♦ s❡rá−E(x)eii= 0✳ ❆ss✐♠✱ ♣❛r❛ t♦❞♦x∈A t❡♠♦s
E(x) =E(x)Idn =E(x)e11+. . .+E(x)enn = 0 +. . .+ 0 = 0.
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡F = 0✳
Pr♦♣♦s✐çã♦ ✷✳✶✳✾✳ ❙❡❥❛C ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❡A =Mn(C)✳ ❊①✐st❡♠ ❢✉♥çõ❡sk✲❛❞✐t✐✈❛s
ζk:Ak →C t❛✐s q✉❡
xn+ζ1(x)xn−1+ζ2(x, x)xn−2+. . .+ζn(x, . . . , x)Idn = 0
♣❛r❛ t♦❞♦ x∈A✳
✷✳✶✳ ❊①❡♠♣❧♦s ❞❡ ■❞❡♥t✐❞❛❞❡s ❋✉♥❝✐♦♥❛✐s
❉❡♠♦♥str❛çã♦✳ P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ ❢❛r❡♠♦s ❝♦♠ ❞❡t❛❧❤❡s ♦ ❝❛s♦ n = 2 ❡ ❞❡♣♦✐s ♦
❝❛s♦ ❣❡r❛❧✳ ❙❡❥❛♠
x=
x11 x12 x21 x22
❡ y=
y11 y12 y21 y22
.
❖ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❞❡x é ❞❛❞♦ ♣♦r
p(λ) = det(λId2−x) = λ2−(x11+x22)λ+ (x11x22−x12x21).
❉❡✜♥✐♥❞♦ζ1 :A→C ❡ζ2 :A2 →C ♣♦r
ζ1(x) = −(x11+x22) ❡ ζ2(x, y) = (x11y22−x12y21),
s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❈❛②❧❡②✲❍❛♠✐❧t♦♥ q✉❡
x2+ζ1(x)x+ζ2(x, x)Id2 = 0
♣❛r❛ t♦❞♦x∈A✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ ❝❛s♦ ❣❡r❛❧ n ≥ 1✳ ❙❡ x = (xij)ij ∈ A = Mn(C)✱ ❡♥tã♦ ♣♦❞❡♠♦s
❡s❝r❡✈❡r s❡✉ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦ ❝♦♠♦
p(λ) = λn+u1(x)λn−1+u2(x)λn−2+. . .+un−1(x)λ+un(x),
♦♥❞❡uk(x) é ❢♦r♠❛❞♦ ♣♦r s♦♠❛s ♦✉ s✉❜tr❛çõ❡s ❞❡ ❡❧❡♠❡♥t♦s ❞♦ t✐♣♦
xi1j1xi2j2. . . xikjk.
❱❛♠♦s ❞❡♥♦t❛r✱ s❡♠ ♠✉✐t♦ r✐❣♦r ♠❛t❡♠át✐❝♦✱
uk(x) =
X
±xi1j1xi2j2. . . xikjk.
❉❡✜♥❛ ❛ ❢✉♥çã♦ ζk : Ak → C ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ s❡ x1 = (xij1)ij, x2 = (x2ij)ij, . . . , xk =
(xk
ij)ij ∈A✱ ❡♥tã♦
ζk(x1, x2, . . . , xk) =
X
±x1i1j1x2i2j2. . . xkikjk.
❚❡♠♦s q✉❡ ζk é k✲❛❞✐t✐✈❛✱ ζk(x, x, . . . , x) = uk(x) ❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈❛②❧❡②✲❍❛♠✐❧t♦♥ ♦
r❡s✉❧t❛❞♦ ❡stá ♣r♦✈❛❞♦✳
