Keidna Cristiane Oliveira Souza

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

■♥✈♦❧✉çõ❡s ❈♦❧♦r✐❞❛s ❡♠ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s

♣♦r

❑❡✐❞♥❛ ❈r✐st✐❛♥❡ ❖❧✐✈❡✐r❛ ❙♦✉③❛

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ■r✐♥❛ ❙✈✐r✐❞♦✈❛

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

■♥✈♦❧✉çõ❡s ❈♦❧♦r✐❞❛s ❡♠ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s

♣♦r

❑❡✐❞♥❛ ❈r✐st✐❛♥❡ ❖❧✐✈❡✐r❛ ❙♦✉③❛

❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❡ss♦r❛ ❉♦✉t♦r❛ ■r✐♥❛ ❙✈✐r✐❞♦✈❛

❉❡❞✐❞♦ ❡st❡ tr❛❜❛❧❤♦ ❛♦s ♠❡✉s ♣❛✐s✱ ❇❡♥✐❝✐♦ ❡ ❆♥❛❀

❆♦s ♠❡✉s ✐r♠ã♦s✱ ❲✐♣s♦♥ ❡ ❑❡❧❧❡❀ ➚ ♠✐♥❤❛ ❛✈ó✱ ❈♦r❛❝✐✳

❙❡♠ ✈♦❝ês ❡✉ ♥❛❞❛ s❡r✐❛✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s ♣♦r ♠❡ ♣r♦♣♦r❝✐♦♥❛r r❡❛❧✐③❛r ♠❛✐s ❡st❡ s♦♥❤♦✳ ❘❡❝♦♥❤❡ç♦✱ ❡♠ t♦❞♦ ♦ ♣❡r❝✉rs♦✱ ❛ s✉❛ ♠ã♦ ❣r❛♥❞✐♦s❛ s♦❜r❡ ♠✐♥❤❛ ✈✐❞❛✳

❆♦ ♠❡✉ s✉♣♦rt❡✱ à ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ❛♣♦✐♦✱ r❡❢❡rê♥❝✐❛✱ ❝❛r✐♥❤♦ ❡ ❝♦♠♣r❡❡♥sã♦✳ ❊s✲ ♣❡❝✐❛❧♠❡♥t❡✱ ❛♦s ♠❡✉s ♣❛✐s✱ ❆♥❛ ❈❛r✈❛❧❤♦ ❡ ❇❡♥✐❝✐♦ ❚❡✐①❡✐r❛✱ à ♠✐♥❤❛ ❛✈ó✱ ❈♦r❛❝✐ ❘♦❞r✐✲ ❣✉❡s✱ ❡ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ❑❡❧❧❡ ❖❧✐✈❡✐r❛ ❡ ❲✐♣s♦♥ ♥❡② ❖❧✐✈❡✐r❛✱ ♣❡❧♦ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ❡st❛♥❞♦ s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♥♦s ❜♦♥s ❡ ♠❛✉s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆ ✈♦❝ês✱ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s ♥ã♦ s❡r✐❛ s✉✜❝✐❡♥t❡ ♣❛r❛ ❡①♣r❡ss❛r ♠❡✉ ❝❛r✐♥❤♦✱ ❛♠♦r ❡ ❣r❛t✐❞ã♦✳

➚ ♠✐♥❤❛ q✉❡r✐❞❛ ♦r✐❡♥t❛❞♦r❛✱ ■r✐♥❛ ❙✈✐r✐❞♦✈❛✱ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠ ❡ t❡r ♠❡ ♣r♦♣♦r❝✐♦♥❛❞♦ t❛♥t❛s ♦♣♦rt✉♥✐❞❛❞❡s✳ P❡❧❛ ❝♦♥✜❛♥ç❛✱ ♣❡❧♦ ❝✉✐❞❛❞♦✱ ♣❡❧❛ ♣r❡♦❝✉♣❛çã♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡♠ r❡s♣♦♥❞❡r ♠✐♥❤❛s ✐♥ú♠❡r❛s ❞ú✈✐❞❛s✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❞✉r❛♥t❡ t♦❞♦ ❡ss❡ t❡♠♣♦✳ ▲❡✈❛r❡✐ ❝♦♠✐❣♦ s❡✉ ❡①❡♠♣❧♦ ❞❡ ♣r♦✜ss✐♦♥❛❧✐s♠♦✳ ❱♦✉ s❡r ♣❛r❛ s❡♠♣r❡ ❣r❛t❛✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ■✈❛♥ ❈❤❡st❛❦♦✈✱ ❉✐♠❛s ❏♦sé ●♦♥ç❛❧✈❡s✱ ◆♦r❛✐ ❘♦♠❡✉ ❘♦❝❝♦ ❡ ❏♦sé ❆♥tô♥✐♦ ❖✳ ❞❡ ❋r❡✐t❛s ♣❡❧❛ ❧❡✐t✉r❛ ❛t❡♥t❛ ❡ ♣❡❧❛s ✈❛❧✐♦s❛s ❝♦rr❡çõ❡s q✉❡ ❡♥r✐q✉❡❝❡r❛♠ ❡st❡ tr❛❜❛❧❤♦✳

➚ ❯❋❚✲❈❛♠♣✉s ❞❡ ❆rr❛✐❛s✱ ❛❣r❛❞❡ç♦ ❛ ❝❛❞❛ ♣r♦❢❡ss♦r q✉❡ ❝♦♥tr✐❜✉✐✉✱ ❞❡ ✉♠❛ ❢♦r♠❛ ♦✉ ❞❡ ♦✉tr❛✱ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦s ♣r♦❢❡ss♦r❡s ❆❞r✐❛♥♦ ❘♦❞r✐❣✉❡s ❡ ❊✉❞❡s ❈♦st❛✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✲❯♥❇✱ ♣❡❧❛s ✐♥ú♠❡r❛s ❡①♣❡r✐ê♥❝✐❛s ❝♦♠♣❛rt✐❧❤❛❞❛s✱ ❛♣♦✐♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❛♠✐③❛❞❡✳ ❊s♣❡❝✐❛❧♠❡♥t❡✱ ❖tt♦✱ ❏♦sé ❈❛r❧♦s✱ ■❧❛♥❛✱ ❑❛❧✐❛♥❛✱ ❊❞✐♠✐❧s♦♥✱ ❊♠❡rs♦♥✱ ▼❛②❡r✱ ❉❛✐❛♥❡✱ ▲❛✐s✱ ❙✉♥❛♠✐t❛✱ ❘❡❣✐❛♥❡✱ ❱❛❧t❡r✱ ❆❧❡①✱ ❇r✉♥♦✱ ❘✐❝❛r❞♦✱ ❈❛♠✐❧❛✱ ●érs✐❝❛ ❡ ❆❧❛♥✳ ❖❜r✐❣❛❞❛ ♣♦r t❡r❡♠ t♦r♥❛❞♦ ❡ss❡ ♣❡rí♦❞♦ ♠❛✐s ❡s♣❡❝✐❛❧✳

❆♦s ♠❡✉s ❛♠✐❣♦s✱ ❆♥á❞r✐❛✱ ●❧á✉❝✐❛✱ ❋❡r♥❛♥❞❛✱ ▼❛r✐❛✱ ❏❛❦❡❧②♥❡✱ ❆❧❡①s❛♥❞r❛✱ ❋❧á✈✐❛ ❡

P❡❞r♦ ❏ú♥✐♦r✱ ♣❡❧❛s ✈❛❧✐♦s❛s ❡ ❝♦♥❢♦rt❛♥t❡s ♣❛❧❛✈r❛s ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝❛r✐♥❤♦ ❞❡ ❛♥♦s✳ ❙❡♠ ✈♦❝ês✱ ❛ ✈✐❞❛ s❡r✐❛ ✉♠ ❞❡s❡rt♦ ❞❡ ❛❧❡❣r✐❛s✳

❆ t♦❞♦s ♦s ♠❡✉s t✐♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ▲❡✈✐✱ ❈❧❡❛✱ ❈❧❡✉③❛✱ ❈❧❛✉❞✐♦✱ ❙❡❧♠❛✱ ❆❞❡❧✐❛✱ ▲✉❝✐❛♥❛ ❡ ❱❛❧❞❡r✱ ♣❡❧♦ ❛♣♦✐♦ ❡ ✐♥❝❡♥t✐✈♦✳

❆♦s ♠❡✉s q✉❡r✐❞♦s ♣r✐♠♦s✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ í♠♣❛r ❞❡s❝♦♥tr❛çã♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛✲❯♥❇✱ ♣❡❧♦ ❛✉①í❧✐♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♣r♦✜ss✐♦♥❛❧ ❡ ♥❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ❈◆Pq✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❘❡s✉♠♦

❙❡❥❛G✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦ ❡ s❡❥❛F ✉♠ ❝♦r♣♦✳ ❙✉♣♦♥❤❛ q✉❡R s❡❥❛ ✉♠ ❛♥❡❧ ✭F✲

á❧❣❡❜r❛✮G✲❣r❛❞✉❛❞♦ ❡σ✉♠2✲❝♦❝✐❝❧♦ ❛♥t✐✲s✐♠étr✐❝♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❝❛r❛❝t❡r✐③❛♠♦s ❛♥é✐s

✭F✲á❧❣❡❜r❛s✮ G✲❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡♠ t❡r♠♦s ❞❡ ♣❛r❡s ❜✐❧✐♥❡❛r❡s ♥ã♦ ❞❡❣❡♥❡r❛❞♦s ❣r❛❞✉❛❞♦s✳ ❙❡G é ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠p✱ ♦♥❞❡pé ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❛♥é✐s ✭F✲á❧❣❡❜r❛s✮G✲❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡ ✉♠❛ σ✲✐♥✈♦❧✉çã♦ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ✉♠❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❤❡r♠✐t✐❛♥❛ ♦✉ ❛♥t✐✲❤❡r♠✐t✐❛♥❛ ❣r❛❞✉❛❞❛✳ ❆❧é♠ ❞❡ ❣❡♥❡r❛❧✐③❛r❡♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❑❛♣❧❛♥s❦② q✉❡ tr❛t❛ ❞❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ ❛♥é✐s ♣r✐♠✐t✐✈♦s✱ ❡ss❡s r❡s✉❧t❛❞♦s t❛♠❜é♠ ❣❡♥❡r❛❧✐③❛♠ ♦s r❡s✉❧t❛❞♦s ❞❡ ❘❛❝✐♥❡✱ ❡♠ ❬✷✺❪✱ ❡ ❇❛❤t✉r✐♥✱ ❇r❡sˇar ❡ ❑♦❝❤❡t♦✈✱ ❡♠ ❬✶❪✱ q✉❡ ❝❧❛ss✐✜❝❛♠ s✉♣❡r✐♥✈♦❧✉çõ❡s ❡♠ s✉♣❡r❛♥é✐s ♣r✐♠✐t✐✈♦s ❡ ✐♥✈♦❧✉çõ❡s ❣r❛❞✉❛❞❛s ❡♠ ❛♥é✐s ❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆✐♥❞❛ ♥♦ ❝❛s♦ ❡♠ q✉❡ G é ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠ ♣r✐♠❛ p✱ ♦❜t❡♠♦s ❝♦r♦❧ár✐♦s r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ✉♠❛ ❞❡s❝r✐çã♦ ❞❡ σ✲✐♥✈♦❧✉çõ❡s ❡♠ á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s s✐♠♣❧❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦❜t❡♠♦s ❞❡s❝r✐çã♦ ❞❡ σ✲✐♥✈♦❧✉çõ❡s ♥♦ ❛♥❡❧ Z3✲❣r❛❞✉❛❞♦ R = Mn(D) ❞❡ ♠❛tr✐③❡s n×n s♦❜r❡ ✉♠

❛♥❡❧ Z3✲❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦D ♥♦ ❝❛s♦ ❞❡ ❛❧❣✉♠❛s ❝❧❛ss❡s ❞❡ ❣r❛❞✉❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❡♠

R✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆♥❡❧ ❣r❛❞✉❛❞♦ ♣r✐♠✐t✐✈♦✱ σ✲✐♥✈♦❧✉çã♦✱ 2✲❝♦❝✐❝❧♦✱ σ✲❛❞❥✉♥t❛ ❡ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦✳

❆❜str❛❝t

▲❡t G ❜❡ ❛ ✜♥✐t❡ ❛❜❡❧✐❛♥ ❣r♦✉♣ ❛♥❞ F ❛ ✜❡❧❞✳ ❙✉♣♣♦s❡ t❤❛t R ✐s ❛ G✲❣r❛❞❡❞ r✐♥❣

✭♦r F✲❛❧❣❡❜r❛✮ ❛♥❞ σ ✐s ❛♥ ❛♥t✐✲s②♠♠❡tr✐❝ ✷✲❝♦❝②❝❧❡✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ❝❤❛r❛❝t❡r✐③❡ r✐❣❤t ♣r✐♠✐t✐✈❡G✲❣r❛❞❡❞ r✐♥❣s ✭F✲❛❧❣❡❜r❛s✮ ✇✐t❤ ❛ ♠✐♥✐♠❛❧ ❣r❛❞❡❞ r✐❣❤t ✐❞❡❛❧ ✐♥ t❡r♠s ♦❢ ♥♦♥✲ ❞❡❣❡♥❡r❛t❡ ❣r❛❞❡❞ ❜✐❧✐♥❡❛r ♣❛✐rs✳ ■❢G✐s ❛ ❣r♦✉♣ ♦❢ ♦r❞❡rp✱ ✇❤❡r❡p✐s ❛ ♣r✐♠❡ ♥✉♠❜❡r✱ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛ r✐❣❤t ♣r✐♠✐t✐✈❡G✲❣r❛❞❡❞ r✐♥❣ ✇✐t❤ ❛ ♠✐♥✐♠❛❧ ❣r❛❞❡❞ r✐❣❤t ✐❞❡❛❧ ❛♥❞ ❛ σ✲✐♥✈♦❧✉t✐♦♥ ✐s r❡❧❛t❡❞ t♦ ❛ ♥♦♥❞❡❣❡♥❡r❛t❡ ǫ✲❤❡r♠✐t✐❛♥ s❡sq✉✐❧✐♥❡❛r ❣r❛❞❡❞ ❢♦r♠✳ ❚❤✐s ❣❡♥❡r❛❧✐s❡s t❤❡ t❤❡♦r❡♠ ♦❢ ❑❛♣❧❛♥s❦② ❛❜♦✉t t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐♥✈♦❧✉t✐♦♥s ✐♥ ♣r✐♠✐t✐✈❡ r✐♥❣s✱ ❛♥❞ s✐♠✐❧❛r r❡s✉❧ts ♦❢ ❘❛❝✐♥❡✱ ✐♥ ❬✷✺❪✱ ❢♦r s✉♣❡r✐♥✈♦❧✉t✐♦♥s✱ ❛♥❞ ♦❢ ❇❛❤t✉r✐♥✱ ❇r❡sˇar✱ ❛♥❞ ❑♦❝❤❡t♦✈✱ ✐♥ ❬✶❪✱ ❢♦r ❣r❛❞❡❞ ✐♥✈♦❧✉t✐♦♥s✳ ❆❧s♦✱ ✇❤❡♥G ✐s ❛ ❣r♦✉♣ ♦❢ ❛ ♣r✐♠❡ ♦r❞❡rp✱ ✇❡ ♦❜t❛✐♥ s♦♠❡ ❝♦r♦❧❧❛r✐❡s ❛❜♦✉t ❞❡s❝r✐♣t✐♦♥ ♦❢σ✲✐♥✈♦❧✉t✐♦♥s ✐♥ s✐♠♣❧❡ ❣r❛❞❡❞ ❛❧❣❡❜r❛s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❞❡s❝r✐❜❡σ✲✐♥✈♦❧✉t✐♦♥s ✐♥ t❤❡ Z3✲❣r❛❞❡❞ r✐♥❣ R=Mn(D) ♦❢n×n ♠❛tr✐❝❡s

♦✈❡r ❛ Z3✲❣r❛❞❡❞ ❞✐✈✐s✐♦♥ r✐♥❣D✱ ❢♦r s♦♠❡ ❝❧❛ss❡s ♦❢ ❡❧❡♠❡♥t❛r② ❣r❛❞✐♥❣s ♦❢R✳

❑❡②✇♦r❞s✿ ●r❛❞❡❞ ♣r✐♠✐t✐✈❡ r✐♥❣✱ σ✲✐♥✈♦❧✉t✐♦♥✱ 2✲❝♦❝②❝❧❡✱ σ✲❛❞❥♦✐♥t ❛♥❞ ❣r❛❞❡❞ ❞✐✲ ✈✐s✐♦♥ r✐♥❣✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✻

✶✳✶ ❆♥é✐s ●r❛❞✉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ▼ó❞✉❧♦s ●r❛❞✉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ ➪❧❣❡❜r❛s ●r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹ ❍♦♠♦♠♦r✜s♠♦s ●r❛❞✉❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✻ ➪❧❣❡❜r❛s ●r❛❞✉❛❞❛s ❙✐♠♣❧❡s ❞❡ ❉✐♠❡♥sã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷ ❘❡s✉❧t❛❞♦s ❆♥t❡❝❡❞❡♥t❡s ✷✼

✷✳✶ ❆♥❡❧ Pr✐♠✐t✐✈♦ ❝♦♠ ■♥✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❙✉♣❡rá❧❣❡❜r❛s Pr✐♠✐t✐✈❛s ❝♦♠ ❙✉♣❡r✐♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ Z3✲✐♥✈♦❧✉çã♦ ♥❛ á❧❣❡❜r❛ Z3✲❣r❛❞✉❛❞❛Mp+q+r(D) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹ ■♥✈♦❧✉çõ❡s ●r❛❞✉❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✺ ➪❧❣❡❜r❛ ❞❡ ❏♦r❞❛♥ ❈♦❧♦r✐❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✸ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s ❝♦♠ σ✲✐♥✈♦❧✉çõ❡s ❡ P❛r ❞❡ ❊s♣❛ç♦s ❉✉❛✐s

●r❛❞✉❛❞♦s ❝♦♠ ❚♦rçã♦ ✹✶

✸✳✶ 2✲❝♦❝✐❝❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✸✳✷ P❛r ❞❡ ❊s♣❛ç♦s ❉✉❛✐s ●r❛❞✉❛❞♦s ❝♦♠ ❚♦rçã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✸✳✸ σ✲✐♥✈♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✹ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s ❡ P❛r❡s ❞❡ ❊s♣❛ç♦s ❉✉❛✐s ●r❛❞✉❛❞♦s ❝♦♠ ❚♦rçã♦ ✻✽

✹ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s ❝♦♠ σ✲✐♥✈♦❧✉çõ❡s ❡ ❋♦r♠❛s ❙❡sq✉✐❧✐♥❡❛r❡s

ǫ✲❤❡r♠✐t✐❛♥❛s ●r❛❞✉❛❞❛s ✼✻

✹✳✶ ❋♦r♠❛ ❙❡sq✉✐❧✐♥❡❛r ●r❛❞✉❛❞❛ ❝♦♠ ❚♦rçã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✷ ❘❡s✉❧t❛❞♦s ❆✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✹✳✸ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✹✳✹ ❆❧❣✉♠❛s ❈♦♥s❡q✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✹✳✺ σ✲✐♥✈♦❧✉çõ❡s ♥♦ ❛♥❡❧ ❞❡ ▼❛tr✐③❡sZ3✲●r❛❞✉❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾

✺ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✶✶✺

■♥tr♦❞✉çã♦

❙❡❥❛G✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦ ❡ s❡❥❛ F ✉♠ ❝♦r♣♦✳ ❈♦♥s✐❞❡r❛♠♦s ❛♥é✐s ❛ss♦❝✐❛t✐✈♦s ❡ F✲á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s✱ ❛♠❜♦s ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡2✳ ❯♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ R é

G✲❣r❛❞✉❛❞♦ s❡ é ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛ ❞✐r❡t❛ ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s ❛❜❡❧✐❛♥♦s ✭F✲s✉❜❡s♣❛ç♦s✮

R=M

α∈G

Rα t❛✐s q✉❡ RαRβ ⊆ Rα+β ♣❛r❛ t♦❞♦sα, β ∈G✳

❙❡❥❛ σ : G×G −→Z ✉♠ 2✲❝♦❝✐❝❧♦ t❛❧ q✉❡ σ(α, β) = σ(β, α)−1 _{♣❛r❛ t♦❞♦s α, β ∈ G✱}

♦♥❞❡ Z = {1,−1} s❡ R é ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦ ❡ Z = F× _{s❡ R é ✉♠❛ F✲á❧❣❡❜r❛ G✲} ❣r❛❞✉❛❞❛✳ ❯♠❛σ✲✐♥✈♦❧✉çã♦ ❡♠ ✉♠ ❛♥❡❧G✲❣r❛❞✉❛❞♦Ré ✉♠❛ ❛♣❧✐❝❛çã♦Z✲❧✐♥❡❛r ❣r❛❞✉❛❞❛ ❞❡ ❣r❛✉ ♥❡✉tr♦ ∗σ :R −→ R q✉❡ s❛t✐s❢❛③ ❛s r❡❧❛çõ❡s

r∗σ∗σ =r ❡ (rαrβ)∗σ =σ(α, β)rβ∗σrα∗σ

♣❛r❛ q✉❛✐sq✉❡r r ∈ R, rα ∈ Rα, rβ ∈ Rβ ❡ α, β ∈ G. ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ✉♠❛ σ✲ ✐♥✈♦❧✉çã♦ ❡♠ ✉♠❛ F✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛ A é ✉♠❛ ❛♣❧✐❝❛çã♦ F✲❧✐♥❡❛r ❣r❛❞✉❛❞❛ ❞❡ ❣r❛✉ ♥❡✉tr♦∗σ :A −→ A t❛❧ q✉❡

a∗σ∗σ =a ❡ (aαaβ)∗σ =σ(α, β)a∗σβ a∗σα ♣❛r❛ q✉❛✐sq✉❡r a∈ A, aα ∈ Aα, aβ ∈ Aβ ❡ α, β ∈G.

❙❡ σ é ✉♠ ❜✐❝❛r❛❝t❡r ❛♥t✐✲s✐♠étr✐❝♦✱ ❞✐③❡♠♦s q✉❡ ∗σ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❝♦❧♦r✐❞❛✳ ❙❡ σ(α, β) = 1 ♣❛r❛ t♦❞♦s α, β ∈ G✱ ❡♥tã♦ ∗σ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❣r❛❞✉❛❞❛✳ ❙❡ G = Z2 ❡

σ(¯α,β¯) = (−1)αβ _{♣❛r❛ α,¯ β}¯ _{∈ G✱ ❡♥tã♦ ∗}

σ é ✉♠❛ s✉♣❡r✐♥✈♦❧✉çã♦✳ P♦r ✜♠✱ s❡ σ(¯α,β¯) =

(−1)αβ_{,❝♦♠ α,¯ β}¯_∈Z

3✱ ∗σ é ✉♠❛ Z3✲✐♥✈♦❧✉çã♦✳

❊♥t❡♥❞❡♠♦s ♣♦r ❛♥❡❧ ❣r❛❞✉❛❞♦ ♣r✐♠✐t✐✈♦ à ❞✐r❡✐t❛ ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦R t❛❧ q✉❡ ❡①✐st❡

✷ ❛♥é✐s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ♠✐♥✐♠❛❧ ❡♠ t❡r♠♦s ❞❡ ❢♦r♠❛s ♥ã♦ ❞❡✲ ❣❡♥❡r❛❞❛s ❤❡r♠✐t✐❛♥❛s ❡ ❛❧t❡r♥❛❞❛s ❬ ❬✷✷❪✱ ❚❤❡♦r❡♠ ✹✳✻✳✽❪✳ ❱❛❧❡ ♠❡♥❝✐♦♥❛r q✉❡ ✉♠❛ ❞❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❚❡♦r❡♠❛ ❞❡ ❑❛♣❧❛♥s❦② é ❛ ❞❡s❝r✐çã♦ ❞❡ ✐♥✈♦❧✉çõ❡s ❞♦ ♣r✐♠❡✐r♦ t✐♣♦ ♥❛ F✲á❧❣❡❜r❛ Mn(F) ❞❛s ♠❛tr✐③❡s n×n s♦❜r❡ ✉♠ ❝♦r♣♦ F ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝✲ t❡ríst✐❝❛0.

▼♦t✐✈❛❞♦ ♣❡❧♦ q✉❡ é ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✻❪✱ ❬✶✾❪✱ ❬✷✷❪ ❡ ❬✷✻❪✱ ❘❛❝✐♥❡ ❡♠ ❬✷✺❪ ♠♦str❛ r❡s✉❧t❛❞♦s ❡str✉t✉r❛✐s ❛♥á❧♦❣♦s ♣❛r❛ s✉♣❡r❛♥é✐s✳ ❊♠ ❬✶✷❪✱ ❱✐❧❧❛ ♣r♦✈❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ s✉✲ ♣❡r✐♥✈♦❧✉çõ❡s ❡♠ s✉♣❡rá❧❣❡❜r❛s ♣r✐♠✐t✐✈❛s✳ ❏á ❡♠ ❬✷✺❪✱ ❘❛❝✐♥❡ t❛♠❜é♠ ❛♣r❡s❡♥t❛ ❞♦✐s t❡♦r❡♠❛s✿ ❬❚❤❡♦r❡♠ ✻❪ ❡ ❬❚❤❡♦r❡♠ ✼❪✳ ◆❡st❡s✱ ❘❛❝✐♥❡ ❝❧❛ss✐✜❝❛ ❛♥é✐s ♣r✐♠✐t✐✈♦s ❝♦♠ ✉♠ s✉♣❡r✐❞❡❛❧ ✉♥✐❧❛t❡r❛❧ ♠✐♥✐♠❛❧ ❡ ❝♦♠ s✉♣❡r✐♥✈♦❧✉çã♦ ❡♠ t❡r♠♦s ❞❡ ❛♣❧✐❝❛çõ❡s ❜✐✲❛❞✐t✐✈❛s ♥ã♦ ❞❡❣❡♥❡r❛❞❛s ❣r❛❞✉❛❞❛s ❡ ❡♠ t❡r♠♦s ❞❡ ❢♦r♠❛s ❤❡r♠✐t✐❛♥❛s ❡ ❛♥t✐✲❤❡r♠✐t✐❛♥❛s ♥ã♦ ❞❡❣❡♥❡r❛❞❛s ❣r❛❞✉❛❞❛s✳ ❊ ♠❛✐s✱ q✉❛♥❞♦ ♦ ❝♦r♣♦F é ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❡ ❞❡ ❝❛r❛❝t❡✲ ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡2✱ ❇❛❤t✉r✐♥✱ ▼✳ ❚✈❛❧❛✈❛❞③❡ ❡ ❚✳ ❚✈❛❧❛✈❛❞③❡ ❞❡s❝r❡✈❡♠ ❛s s✉♣❡rá❧❣❡❜r❛s

s✐♠♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ❝♦♠ s✉♣❡r✐♥✈♦❧✉çã♦ ❬✸❪✳

❏❛❜❡r ❡♠ ❬✶✽❪ ❡st✉❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ Z3✲✐♥✈♦❧✉çõ❡s ♥❛ á❧❣❡❜r❛ Z3✲❣r❛❞✉❛❞❛ A =

Mp+q+p(D), ♦♥❞❡ D é ✉♠❛ á❧❣❡❜r❛ ❞❡ ❞✐✈✐sã♦✳

❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦ ❡ F ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡✲ ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡2✳ ❊♠ ❬✶❪✱ ❇❛❤t✉r✐♥✱ ❇r❡sˇar ❡ ❑♦❝❤❡t♦✈✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❝❧❛ss✐✜❝❛r✱

❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ t♦❞❛s ❛s G✲❣r❛❞✉❛çõ❡s ❞❛ F✲á❧❣❡❜r❛ ❞❡ ▲✐❡ ✜♥✐tár✐❛ s✐♠♣❧❡s ❞❡ tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s ✭❧✐♥❡❛r ❡s♣❡❝✐❛❧✱ ♦rt♦❣♦♥❛❧ ❡ s✐♠♣❧ét✐❝❛✮ ❡♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ F✱ ♣r♦✈❛r❛♠ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ F✲á❧❣❡❜r❛s ✭♦✉ ❛♥é✐s✮G✲❣r❛❞✉❛❞❛s ♣r✐♠✐t✐✈❛s à ❡sq✉❡r❞❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❡sq✉❡r❞❛G✲❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡ ✐♥✈♦❧✉çã♦ ❣r❛❞✉❛❞❛✱ s✐♠✐❧❛r ❛♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✷✺❪✳ ❊♠ ❬✷✼❪✱ ❙✈✐r✐❞♦✈❛ ❞❡s❝r❡✈❡ t♦❞❛s ❛s á❧❣❡❜r❛s∗gr✲❣r❛❞✉❛❞❛s s✐♠♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ ♦♥❞❡G=Zq✱qé ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ♦✉ q= 4✱ F é ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ ❡∗gr é ✉♠❛ ✐♥✈♦❧✉çã♦ ❣r❛❞✉❛❞❛✳ ❚❛♠❜é♠ ♣❛r❛ ✐♥✈♦❧✉çõ❡s ❣r❛❞✉❛❞❛s ❇❛❤t✉r✐♥✱ ❙❤❡st❛❦♦✈ ❡ ❩❛✐❝❡✈✱ ❡♠ ❬✹❪✱ ❡ ❇❛❤t✉r✐♥ ❡ ❩❛✐❝❡✈✱ ❡♠ ❬✺❪✱ ❞❡s❝r❡✈❡♠ ❛s ✐♥✈♦❧✉çõ❡s ❣r❛❞✉❛❞❛s ❡♠Mn(F)q✉❛♥❞♦ F é ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2✳ ❏á ❇❛❤t✉r✐♥ ❡ ●✐❛♠❜r✉♥♦

❡♠ ❬✷❪ ❞❡s❝r❡✈❡♠ G✲❣r❛❞✉❛çã♦ ❡♠ Mn(F) ❛❞♠✐t✐♥❞♦ ✉♠❛ ✐♥✈♦❧✉çã♦ ❣r❛❞✉❛❞❛✱ t❛♠❜é♠

♣❛r❛ F ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦ ❡ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2✳

❇❡r❣❡♥ ❡ ●r③❡s③❝③✉❦ ♠♦str❛♠ ❡♠ ❬✾❪ ❝♦♠♦ á❧❣❡❜r❛s ❞❡ ❏♦r❞❛♥ ❝♦❧♦r✐❞❛s s✐♠♣❧❡s s✉r✲ ❣❡♠ ♥❛t✉r❛❧♠❡♥t❡ ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❣r❛❞✉❛❞❛s s✐♠♣❧❡s ❡ ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❣r❛❞✉❛❞❛s s✐♠♣❧❡s ❝♦♠ ✐♥✈♦❧✉çã♦ ❝♦❧♦r✐❞❛✳

✸ ❡ss❛ t❡♦r✐❛ ❞❡s❡♥✈♦❧✈✐❞❛ ❡♠ ❬✶❪✱ ❬✶✷❪✱ ❬✷✷❪ ❡ ❬✷✺❪✱ ♥♦ss♦ tr❛❜❛❧❤♦ s❡❣✉✐rá ♥❡ss❛ ❧✐♥❤❛✳ ❊s✲ t✉❞❛♠♦s ❛ ❡str✉t✉r❛ ❞❡ ❛♥é✐sG✲❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❡F✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s ♣r✐♠✐t✐✈❛s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ G✲❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧✳ ❚r❛❜❛❧❤❛♠♦s ❝♦♠ ❞♦✐s t✐♣♦s ❞❡ ❛♣❧✐❝❛çõ❡s ❣r❛❞✉❛❞❛s✿ ♣❛r ❜✐❧✐♥❡❛r ♥ã♦ ❞❡❣❡♥❡r❛❞♦ ❣r❛❞✉❛❞♦ ❡ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❣r❛❞✉❛❞❛✳ ❉❡✜♥✐♠♦s ❛ σ✲❛❞❥✉♥t❛ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❤♦♠♦❣ê♥❡♦ ❞♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❞❡ ❡♥❞♦♠♦r✜s♠♦s ❞❡ ✉♠ ♠ó❞✉❧♦ ❣r❛❞✉❛❞♦✳

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❝❧❛ss✐✜❝❛r ❛♥é✐s ✭F✲á❧❣❡❜r❛s✮ G✲❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ G✲❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡♠ t❡r♠♦s ❞❡ ♣❛r❡s ❜✐❧✐♥❡❛r❡s ♥ã♦ ❞❡❣❡♥❡r❛❞♦s ❣r❛❞✉❛❞♦s ❡ ❛♥é✐s ❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡ ❝♦♠σ✲✐♥✈♦❧✉çã♦ ❡♠ t❡r♠♦s ❞❡ ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡s ♥ã♦ ❞❡❣❡♥❡r❛❞❛s ❣r❛❞✉❛❞❛s✱ ❝♦♠♦ ❢❡✐t♦ ❡♠ ❬✷✷❪✳ ◆♦ ♣r✐♠❡✐r♦ t❡♦r❡♠❛ ❡①✐❣✐♠♦s ❝♦♠♦ ❤✐♣ót❡s❡ q✉❡ G s❡❥❛ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦ ❡ ♥♦ s❡❣✉♥❞♦ t❡♦r❡♠❛ q✉❡Gs❡❥❛ ✉♠ ❣r✉♣♦ ❞❡ ♦r❞❡♠ p✱ ♦♥❞❡p é ✉♠ ♥ú♠❡r♦ ♣r✐♠♦✳

❙❡❥❛♠ D ✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ G✲❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦✱ V ✉♠ D✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ à ❡s✲

q✉❡r❞❛ G✲❣r❛❞✉❛❞♦ ❡ W ✉♠ D✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ à ❞✐r❡✐t❛ G✲❣r❛❞✉❛❞♦✱ h−,−iν ✉♠ ♣❛r ❜✐❧✐♥❡❛r ♥ã♦ ❞❡❣❡♥❡r❛❞♦ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦ ❛♦ ♣❛r ❞❡ ❡s♣❛ç♦s ❞✉❛✐s ❝♦♠ t♦rçã♦V ×W ❡

∗σ ❛ σ✲❛❞❥✉♥t❛ ❛ss♦❝✐❛❞❛ ❛ h−,−iν ✭✈❡❥❛ ❉❡✜♥✐çã♦ ✸✳✷✳✶ ❡ ❉❡✜♥✐çã♦ ✸✳✷✳✸✮✳ ❉❡♥♦t❛✲ ♠♦s ♣♦rLgrσ_W (V)♦ s✉❜❛♥❡❧ ✭F✲s✉❜á❧❣❡❜r❛✮ ❣r❛❞✉❛❞♦ ❞❡Endgr_D(V)❞❡ t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s

q✉❡ ♣♦ss✉❡♠σ✲❛❞❥✉♥t❛ ❡ F_Wgrσ(V)♦ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❣r❛❞✉❛❞♦ ❞❡Lgrσ_W (V) ❞❡ t♦❞♦s ♦s ♦♣❡✲

r❛❞♦r❡s q✉❡ ♣♦ss✉❡♠ ♣♦st♦ ✜♥✐t♦ ❡ ♣♦ss✉❡♠ σ✲❛❞❥✉♥t❛✳ ❆s ❝❧❛ss✐✜❝❛çõ❡s q✉❡ ♦❜t✐✈❡♠♦s ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠LgrσW (V) ❡F

grσ

W (V)✳ ❆ s❛❜❡r✱ ♣r♦✈❛♠♦s ♦s✿

❚❡♦r❡♠❛ ✸✳✹✳✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦✳ ❙❡ R é ✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ G✲ ❣r❛❞✉❛❞♦ ♣r✐♠✐t✐✈♦ à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡σ é ✉♠ 2✲❝♦❝✐❝❧♦

❛♥t✐✲s✐♠étr✐❝♦ t❛❧ q✉❡ σ(α,−α)2 _{= 1 ♣❛r❛ t♦❞♦ α ∈ G, ❡♥tã♦ ❡①✐st❡ ✉♠ ♣❛r ❞❡ ❡s♣❛ç♦s}

❞✉❛✐s ❝♦♠ t♦rçã♦ DV ❡ WD t❛❧ q✉❡

F_Wgrσ(V)⊆ R ⊆ Lgrσ_W (V),

♦♥❞❡ D é ✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ G✲❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ❞❛❞♦ ✉♠ ♣❛r ❞❡ ❡s♣❛ç♦s ❞✉❛✐s ❝♦♠ t♦rçã♦ V ❡ W s♦❜r❡ ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦ D✱ q✉❛❧q✉❡r ❛♥❡❧

G✲❣r❛❞✉❛❞♦ R s❛t✐s❢❛③❡♥❞♦

FWgrσ(V)⊆ R ⊆ L grσ W (V)

é ❣r❛❞✉❛❞♦ ♣r✐♠✐t✐✈♦ à ❞✐r❡✐t❛ ❡ R ❝♦♥té♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧✳ ❆❧é♠

❞✐ss♦✱ FWgrσ(V) é ♦ ú♥✐❝♦ ✐❞❡❛❧ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❞❡ R✳

✹ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡ ✉♠❛ σ✲✐♥✈♦❧✉çã♦ ⋆σ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛

❣r❛❞✉❛❞♦ V t❛❧ q✉❡✿

❛✮ V ×V é ✉♠ ♣❛r s❡sq✉✐❧✐♥❡❛r à ❡sq✉❡r❞❛ ❝♦♠ t♦rçã♦❀ ❜✮ Endgr_R(V) ♣♦ss✉✐ ✉♠❛ σ✲✐♥✈♦❧✉çã♦❀

❝✮ ⋆σ é ❛σ✲❛❞❥✉♥t❛ ❛ss♦❝✐❛❞❛ ❛ ✉♠❛ ❢♦r♠❛ s❡sq✉✐❧✐♥❡❛r ❤❡r♠✐t✐❛♥❛ ♦✉ ❛♥t✐✲❤❡r♠✐t✐❛♥❛ ♥ã♦ ❞❡❣❡♥❡r❛❞❛ ❣r❛❞✉❛❞❛❀

❞✮ FVgr ⊆ R ⊆ L gr

V ❡ R é ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❛çã♦ ❞❡ ⋆σ✳

❖ ❚❡♦r❡♠❛ ✸✳✹✳✷ ❣❡♥❡r❛❧✐③❛ ♦s t❡♦r❡♠❛s ❬❬✷✺❪✱ ❚❤❡♦r❡♠ ✻❪ ❡ ❬❬✶❪✱ ❚❤❡♦r❡♠ ✸✳✸❪✱ ♦s q✉❛✐s ♦s ❞♦✐s ú❧t✐♠♦s s❡ r❡❛❧✐③❛♠ ❡♠σ(¯α,β¯) = (−1)αβ _{♣❛r❛ t♦❞♦sα,¯ β}¯_∈Z

2 ❡σ(α, β) = 1

♣❛r❛ t♦❞♦s α, β ∈ G✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦ ❚❡♦r❡♠❛ ✸✳✹✳✷ ❝❛r❛❝t❡r✐③❛♠♦s ❛♥é✐s ❣r❛❞✉✲

❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡♠ t❡r♠♦s ❞❡ ♣❛r❡s ❜✐❧✐♥❡❛r❡s ♥ã♦ ❞❡❣❡♥❡r❛❞♦s ❣r❛❞✉❛❞♦s✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ♥❡ss❡ t❡♦r❡♠❛ ♥ã♦ ♣❡✲ ❞✐♠♦s q✉❡ ❡①✐st❛ ✉♠❛σ✲✐♥✈♦❧✉çã♦ ♥❛ F✲á❧❣❡❜r❛ ✭❛♥❡❧✮ G✲❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐sã♦✱ ♣♦✐s s❡D

é ✉♠❛F✲á❧❣❡❜r❛G✲❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐sã♦ ♣♦❞❡ ❡①✐st✐r ✉♠2✲❝♦❝✐❝❧♦σ t❛❧ q✉❡D ♥ã♦ ❛❞♠✐t❡

✉♠❛ σ✲✐♥✈♦❧✉çã♦✳ ❆ss✐♠ é ♦ ❝❛s♦ ❞❛ á❧❣❡❜r❛ ❞❡ ❞✐✈✐sã♦ Z2✲❣r❛❞✉❛❞❛ F[Z2]✱ ✈✐st♦ q✉❡

F[Z2] ♥ã♦ ❛❞♠✐t❡ s✉♣❡r✐♥✈♦❧✉çõ❡s ✭✈❡❥❛ ❬✶✹❪✮✳

❏á ♦ ❚❡♦r❡♠❛ ✹✳✸✳✶ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ t❡♦r❡♠❛ ❬❬✷✺❪✱ ❚❤❡♦r❡♠ ✼❪✳ ◆❡ss❡ ❝❛r❛❝t❡r✐③❛♠♦s ❛♥é✐s ❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s à ❞✐r❡✐t❛ ❝♦♠ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ♠✐♥✐♠❛❧ ❡σ✲✐♥✈♦❧✉çõ❡s ❡♠ t❡r♠♦s ❞❡ ❢♦r♠❛s s❡sq✉✐❧✐♥❡❛r❡sǫ✲❤❡r♠✐t✐❛♥❛s ♥ã♦ ❞❡❣❡♥❡r❛❞❛s ❣r❛❞✉❛❞❛s✳ ▼♦str❛♠♦s ❛✐♥❞❛ q✉❡ ♦ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ ❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦ D ❛❞♠✐t❡ ✉♠❛

σ✲✐♥✈♦❧✉çã♦✳

❱ár✐❛s ❝♦♥s❡q✉ê♥❝✐❛s ❡ ❛♣❧✐❝❛çõ❡s sã♦ ♦❜t✐❞❛s ❞❡ t♦❞♦s ❡ss❡s r❡s✉❧t❛❞♦s✳

❈❛♣ít✉❧♦

1

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛❞♦s à t❡♦r✐❛ ❞❡ ❛♥é✐s ❛ss♦❝✐❛t✐✈♦s ❣r❛❞✉❛❞♦s ❡ à t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❛s ❣r❛❞✉❛❞❛s ♣♦r ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ✜♥✐t♦s✱ ♦s q✉❛✐s ❢❛③❡♠ ♣❛rt❡ ❞❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ♥❡❝❡ssár✐❛ às ❞✐s❝✉ssõ❡s ❞❡st❡ tr❛❜❛❧❤♦✳ ❊ss❡ ❛♣❛r❛t♦ t❡ór✐❝♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✻❪✱ ❬✼❪✱ ❬✽❪✱ ❬✶✶❪ ❡ ❬✷✸❪✳ ❊♠ t♦❞♦ tr❛❜❛❧❤♦✱ Gé ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛❞✐t✐✈♦ ✜♥✐t♦✳

✶✳✶ ❆♥é✐s ●r❛❞✉❛❞♦s

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛q✉✐ é ❛♣r❡s❡♥t❛r ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ ❛♥é✐s ❣r❛✲ ❞✉❛❞♦s✳

❙❡❥❛(R,+, .)✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦ ❡ s❡❥❛(G,+)✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦ ❝♦♠ ❡❧❡♠❡♥t♦

♥❡✉tr♦0✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ ❛♥❡❧ R é ❝❤❛♠❛❞♦ G✲❣r❛❞✉❛❞♦ s❡ R ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ s♦♠❛

❞✐r❡t❛ ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s Rα

R=M

α∈G

Rα ✭✶✳✶✮

t❛✐s q✉❡ RαRβ ⊆ Rα+β ♣❛r❛ t♦❞♦s α, β ∈G✳

❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ G=Z2✱ ❞✐③❡♠♦s q✉❡ R é ✉♠ s✉♣❡r❛♥❡❧✳

❆ ❣r❛❞✉❛çã♦ é ❞✐t❛ ♥ã♦ tr✐✈✐❛❧ s❡Rα 6= (0)♣❛r❛ ❛❧❣✉♠ 06=α ∈G. ❖s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ h(R) = [

α∈G

✼ ◗✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ r ∈ R é ❡①♣r❡ss♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ s♦♠❛ ✜♥✐t❛ ❞❡

❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s✿ r =X

α∈G

rα✱ ♦♥❞❡ rα ∈ Rα. ❖s ❡❧❡♠❡♥t♦s ♥ã♦ ♥✉❧♦s rα ♥❛ ❞❡❝♦♠✲ ♣♦s✐çã♦ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠♣♦♥❡♥t❡s ❤♦♠♦❣ê♥❡❛s ❞❡r✳

❙❡ X é ✉♠ s✉❜❛♥❡❧ ♥ã♦ ♥✉❧♦ ❞❡ R✱ ❡♥tã♦ ❡s❝r❡✈❡♠♦s Xα = X ∩ Rα ♣❛r❛ α ∈ G✳ ❉✐③❡♠♦s q✉❡X é ✉♠ s✉❜❛♥❡❧ ❣r❛❞✉❛❞♦ ❞❡ R s❡X =M

α∈G

Xα✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜t❡♠♦s ❛s

♥♦t❛çõ❡s ❡ ♥♦çõ❡s ❞❡ ✐❞❡❛❧ à ❡sq✉❡r❞❛ ❣r❛❞✉❛❞♦ ✱ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❡ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❣r❛❞✉❛❞♦ q✉❛♥❞♦ X é✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✉♠ ✐❞❡❛❧ à ❡sq✉❡r❞❛✱ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❡ ✉♠

✐❞❡❛❧ ❜✐❧❛t❡r❛❧✳ ❖❜s❡r✈❛♠♦s q✉❡ ✉♠ s✉❜❛♥❡❧ ✭♦✉ ✐❞❡❛❧✮ é ❣r❛❞✉❛❞♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ é ❣❡r❛❞♦ ❝♦♠♦ ❛♥❡❧ ✭✐❞❡❛❧✮ ♣♦r ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ I é ✉♠ ✐❞❡❛❧

❜✐❧❛t❡r❛❧ ❣r❛❞✉❛❞♦ ❞❡R,♦ q✉♦❝✐❡♥t❡ R/I é ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❝♦♠ ❣r❛❞✉❛çã♦ ❞❛❞❛ ♣♦r✿

(R/I)α := (Rα+I)/I ❡ R/I =

M

α∈G

(R/I)α✳

❊①❡♠♣❧♦ ✶✳✶✳✶✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦ ❡ S =R[x1, . . . , xd] ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s

❝♦♠✉t❛t✐✈♦s ❡ ❛ss♦❝✐❛t✐✈♦s ♥❛s ✈❛r✐á✈❡✐s x1, . . . , xd✳ ❉❛❞♦ ✉♠❛ d✲✉♣❧❛ (α1, . . . , αd)∈ Zd✱

♦ ❛♥❡❧S é ♠✉♥✐❞♦ ❝♦♠ ❛ s❡❣✉✐♥t❡Z✲❣r❛❞✉❛çã♦✿ S =M

n∈Z Sn,

♦♥❞❡

Sn=

  

X

(m1,...,md)= ¯m∈Zd

rmx¯ m1 1. . . x

md

d | rm¯ ∈R, α1m1+. . .+αdmd=n

  .

❖❜s❡r✈❛♠♦s q✉❡ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ Gé ✉♠ ❣r✉♣♦ ✐♥✜♥✐t♦✳

❉❡ ♣♦ss❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛♥❡❧ ❣r❛❞✉❛❞♦✱ t❡♠♦s ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❜❡♠ ❝♦♥❤❡❝✐❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ✭✈❡❥❛ ❬❬✼❪✱ Pr♦♣♦s✐t✐♦♥ ✶❪ ❡ ❬❬✷✸❪✱ Pr♦♣♦s✐t✐♦♥ ✶✳✶✳✶❪✮✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✳ ❙❡❥❛ R = M

α∈G

Rα ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦ ✉♥✐tár✐♦✳ ❊♥tã♦✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

❛✮ 1∈ R0 ❡ R0 é ✉♠ s✉❜❛♥❡❧ ❞❡ R;

❜✮ s❡ r−1 _{é ♦ ✐♥✈❡rs♦ ❞♦ ❡❧❡♠❡♥t♦ ❤♦♠♦❣ê♥❡♦ r ∈ Rα, ❡♥tã♦ r}−1 _{∈ R−α✳}

❉❡♠♦♥str❛çã♦✳ ❛✮ ❈♦♠♦ R0R0 ⊆ R0 ❡ R0 é ✉♠ s✉❜❣r✉♣♦ ❞❡ R✱ ❜❛st❛ ♠♦str❛r q✉❡

1 ∈ R0. ❙❡❥❛ 1 =

X

α∈G

rα ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ 1 ❝♦♠ rα ∈ Rα. ❊♥tã♦✱ ♣❛r❛ t♦❞♦ sβ ∈ Rβ✱ t❡♠♦s q✉❡

sβ =sβ1 = X

α∈G

✽ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱sβrα = 0♣❛r❛ t♦❞♦α6= 0✳ ❆ss✐♠✱ ♣❛r❛ q✉❛❧q✉❡rs∈ R✱srα= 0

♣❛r❛ t♦❞♦ α 6= 0✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ s = 1✱ ♦❜t❡♠♦s rα = 0 ♣❛r❛ q✉❛❧q✉❡r α6= 0. P♦rt❛♥t♦✱ 1 =r0 ∈ R0 ❡✱ ❝♦♠ ✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ R0 é ✉♠ s✉❜❛♥❡❧ ❞❡R.

❜✮ ❆ss✉♠❛ q✉❡ r ∈ Rλ é ✐♥✈❡rtí✈❡❧✳ ❙❡ r−1 =

X

α∈G

(r−1)α✱ ♦♥❞❡ (r−1₎

α ∈ Rα✱ ❡♥tã♦

1 = rr−1 ₌ X

α∈G

r(r−1)α. ❏á q✉❡ 1 ∈ R0 ❡ r(r−1)α ∈ Rλ+α✱ t❡♠♦s q✉❡ r(r−1)α = 0 ♣❛r❛ t♦❞♦ α 6= −λ. ❈♦♠♦ r é ✐♥✈❡rtí✈❡❧✱ s❡❣✉❡ q✉❡ (r−1₎

α 6= 0 ♣❛r❛ α = −λ✳ P♦rt❛♥t♦✱ r−1 _{= (r}−1₎

−λ ∈ R−λ. ❊ ✐st♦ ✜♥❛❧✐③❛ ❛ ♣r♦✈❛ ❞❛ ♣r♦♣♦s✐çã♦✳

❱❛❧❡ r❡ss❛❧t❛r q✉❡ Rα é ✉♠ R0✲❜✐♠ó❞✉❧♦ ♣❛r❛ t♦❞♦ α∈G✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❯♠ ❛♥❡❧ ✉♥✐tár✐♦ G✲❣r❛❞✉❛❞♦ é ❞❡♥♦♠✐♥❛❞♦ ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦ s❡ t♦❞♦s ♦s s❡✉s ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ♥ã♦ ♥✉❧♦s sã♦ ✐♥✈❡rtí✈❡✐s✳

❙❡R é ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦✱ ❝❧❛r❛♠❡♥t❡ R0 é ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦✳ ❖ ❛♥❡❧ ♦♣♦st♦ G✲❣r❛❞✉❛❞♦ ❞❡ R✱ Ropgr_✱ é ♦ ❣r✉♣♦ ❛❞✐t✐✈♦ ❣r❛❞✉❛❞♦ R ❝♦♠ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r

rα◦opgr rβ :=rβrα ✭✶✳✷✮

♣❛r❛ q✉❛✐sq✉❡r rα ∈ Rα, rβ ∈ Rβ ❡ α, β ∈G.

❋✐♥❛❧✐③❛♠♦s ❡st❛ s❡çã♦ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s ❣r❛❞✉❛❞♦s✱ q✉❡ é s✐♠✐❧❛r à ❞❡✜♥✐çã♦ ❞❡ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❙❡❥❛♠ R = M

α∈G

Rα ❡ B =

M

α∈G

Bα ❞♦✐s ❛♥é✐s G✲❣r❛❞✉❛❞♦s✳ ❯♠ ❤♦✲ ♠♦♠♦r✜s♠♦ ✭✐s♦♠♦r✜s♠♦✮ ❞❡ ❛♥é✐s φ :R −→ B é ❝❤❛♠❛❞♦ ❞❡ ❤♦♠♦♠♦r✜s♠♦ ✭✐s♦♠♦r✲

✜s♠♦✮ ❞❡ ❛♥é✐s ❣r❛❞✉❛❞♦s s❡ φ ♣r❡s❡r✈❛ ❛ ❡str✉t✉r❛ ❣r❛❞✉❛❞❛✱ ✐st♦ é✱ φ(Rα) ⊆ Bα ♣❛r❛ t♦❞♦ α∈G.

❙❡❥❛♠ R =M

α∈G

Rα ❡ B =

M

α∈G

Bα ❛♥é✐s G✲❣r❛❞✉❛❞♦s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ ❛❞✐t✐✈❛ ❞❡ ❛♥é✐s ϕ:R −→ B é ❝❤❛♠❛❞❛ ❞❡ ❣r❛❞✉❛❞❛ ❞❡ ❣r❛✉ ✭❤♦♠♦❣ê♥❡♦✮β s❡ϕ(Rα)⊆ Bα+β ♣❛r❛ t♦❞♦ α∈G.

✾

✶✳✷ ▼ó❞✉❧♦s ●r❛❞✉❛❞♦s

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ ✉♠ ✐♠♣♦rt❛♥t❡ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✱ ❛ s❛❜❡r✱ ♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s✳ ❖ ❡st✉❞♦ ❞❡ t❛❧ ♦❜❥❡t♦ ♠♦t✐✈♦✉ ✐♥ú♠❡r♦s ❛✈❛♥ç♦s ♥❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s✱ t❛❧ ❝♦♠♦ ❛ ❝❧❛ss✐✜❝❛çã♦ ❞❡ á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s s✐♠♣❧❡s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ ✉♠ ❝♦r♣♦ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❢❡❝❤❛❞♦✳

❙❡❥❛R ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦✳ ❈♦♥s✐❞❡r❡(M,+) ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ M é ❝❤❛♠❛❞♦ G✲❣r❛❞✉❛❞♦ s❡ M =M

α∈G Mα,

♦♥❞❡ {Mα | α ∈ G} é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s ❞♦ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ (M,+) ❡

mαrβ ∈Mα+β ♣❛r❛ q✉❛✐sq✉❡r rβ ∈ Rβ, mα ∈Mα ❡ α, β ∈G.

❖s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ h(M) = [

α∈G

Mα sã♦ ❝❤❛♠❛❞♦s ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ❞♦ ♠ó❞✉❧♦M✱ ❡ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦m =X

α∈G

mα é ❞❡❝♦♠♣♦st♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ s♦♠❛ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦smα✳

❯♠ R✲s✉❜♠ó❞✉❧♦ N ❞❡ ✉♠R✲♠ó❞✉❧♦G✲❣r❛❞✉❛❞♦ M éG✲❣r❛❞✉❛❞♦ s❡ N =M

α∈G Nα,

♦♥❞❡ Nα =N ∩Mα✱ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ♣❛r❛ ❝❛❞❛ n ∈ N✱ t♦❞❛s ❛s s✉❛s ❝♦♠♣♦♥❡♥t❡s ❤♦♠♦❣ê♥❡❛s t❛♠❜é♠ ♣❡rt❡♥❝❡♠ ❛N✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛G✲❣r❛❞✉❛❞♦ é ✉♠

R✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦✳

❯♠R✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦M ✐♥❞✉③ ✉♠❛ ❣r❛❞✉❛çã♦ ♥♦ R✲♠ó❞✉❧♦ q✉♦❝✐❡♥t❡M/N✱ ♦♥❞❡ N é ✉♠ s✉❜♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ ❞❡M✳ ❚❛❧ ❣r❛❞✉❛çã♦ é ❞❛❞❛ ♣♦r✿ (M/N)α :={m+N |m∈ Mα}. ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ G✲❣r❛❞✉❛❞♦ ❡ R✲❜✐♠ó❞✉❧♦

G✲❣r❛❞✉❛❞♦✳ ❊♠ ❣❡r❛❧✱ t♦❞♦ ♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ M ❝♦♥té♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s s✉❜♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s✱ M ❡ {0}✱ ♦s q✉❛✐s sã♦ ❝❤❛♠❛❞♦s tr✐✈✐❛✐s✳

❙❡❥❛ M ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦✳ ➱ ❜❡♠ ❝♦♥❤❡❝✐❞♦ q✉❡ ♦ ❛♥✉❧❛❞♦r à ❞✐r❡✐t❛

❞❡ ✉♠R✲♠ó❞✉❧♦ M✱ Annr

R(M) ={r∈ R | mr = 0, ∀ m∈M}✱ é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡

R.❖ ❧❡♠❛ ❛❜❛✐①♦ ♠♦str❛ q✉❡ Annr

R(M)❤❡r❞❛ ❛ ❡str✉t✉r❛ ❣r❛❞✉❛❞❛ ❞❡ R✳ ▲❡♠❛ ✶✳✷✳✶✳ ❙❡❥❛ M ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦✳ ❊♥tã♦✱ Annr

✶✵ ❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝❛❞❛ α ∈ G✱ ❝♦♥s✐❞❡r❡ Annr

R(M)α ={rα ∈ Rα | mrα = 0, ∀ m ∈ M}✳ Pr✐♠❡✐r❛♠❡♥t❡ ♦❜s❡r✈❛♠♦s q✉❡ Annr

R(M)α é ✉♠ s✉❜❣r✉♣♦ ❞❡ Annr_R(M) ♣❛r❛ t♦❞♦ α∈G✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡r ∈Annr

R(M)α∩

X

α6=β∈G

Annr_R(M)β✱ ❡♥tã♦ r∈ Rα∩

X

α6=β∈G

Rβ.

❈♦♠♦Rα∩

X

α6=β∈G

Rβ = (0)✱ t❡♠♦s q✉❡r = 0. ❈♦♠ ✐ss♦✱ ❛ s♦♠❛

X

α∈G

Annr_R(M)α é ❞✐r❡t❛✳ ❙❡❥❛ r ∈ Annr

R(M)✱ ❡♥tã♦ r =

X

α∈G

rα, ♦♥❞❡ rα ∈ Rα ♣❛r❛ t♦❞♦ α ∈ G, ❡ mr = 0 ♣❛r❛ t♦❞♦ m ∈M✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ 0 = mβr = X

α∈G

mβrα ♣❛r❛ t♦❞♦ β ∈ G✳ ❆ss✐♠✱ ❝♦♠♦ M é ❣r❛❞✉❛❞♦✱ t❡♠♦s q✉❡mβrα = 0 ♣❛r❛ t♦❞♦α ∈G✳ ▲♦❣♦✱ Annr

R(M)⊆

M

α∈G Annr

R(M)α. ❊✱ ♣♦rt❛♥t♦✱ Annr

R(M) =

M

α∈G

Annr_R(M)α.

❆♥❛❧♦❣❛♠❡♥t❡✱ ♦ ❛♥✉❧❛❞♦r à ❡sq✉❡r❞❛ ❞❡ ✉♠ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛ J✱ Annl

R(J) =

{r∈ R | rj= 0, ∀j ∈J}✱ é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❣r❛❞✉❛❞♦ ❞❡ R.

❊♠ ♣❛rt✐❝✉❧❛r✱ ♣❛r❛ ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ R ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ❛♥✉❧❛❞♦r à ❡sq✉❡r❞❛

Annl

R(RR) ❡ ♦ ❛♥✉❧❛❞♦r à ❞✐r❡✐t❛ AnnrR(RR)✳ ❆♠❜♦s sã♦ ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s ❣r❛❞✉❛❞♦s ❞❡

R.

❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❯♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ M é ❞✐t♦ ✜❡❧ s❡ Annr

R(M) = (0)✳ ❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❯♠R✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ M é ❞✐t♦ ✐rr❡❞✉tí✈❡❧ s❡ 0 ❡ M sã♦ s❡✉s ú♥✐❝♦s

R✲s✉❜♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s✳

❙❡❥❛K ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ❞♦ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛G✲❣r❛❞✉❛❞♦M✳ ❊♥tã♦✱K = [

β∈G

Kβ✱ ♦♥❞❡Kβ é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡K ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ❞❡

❣r❛✉ β∈G. ❙❡❥❛I ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❞❡ R✳ ❉❡♥♦t❛♠♦s KI ={

m_Xl,nj

l,j

kjil | kj ∈

K, il∈I, ml, nj ∈N }✳

▲❡♠❛ ✶✳✷✳✷✳ ❙❡❥❛♠ M ✉♠R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❡ I ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❞❡ R✳ ❙❡ K ⊆ M é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ♦✉ ✉♠ s✉❜♠ó❞✉❧♦ ❣r❛❞✉❛❞♦✱ ❡♥tã♦ KI é ✉♠ R✲s✉❜♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ ❞❡ M✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ I é ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❡ K ⊆ M✱ t❡♠♦s q✉❡ KI ⊆ M ❡ KI é ✉♠ R✲s✉❜♠ó❞✉❧♦ ❞❡ M. ❱❛♠♦s ♠♦str❛r q✉❡ é ❣r❛❞✉❛❞♦✳ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ s✉❜❣r✉♣♦ ❞❡ M

(KI)τ =

X

✶✶ ❈♦♠♦ M é ❣r❛❞✉❛❞♦ ❡ (KI)τ ⊆ Mτ✱ t❡♠♦s q✉❡ (KI)τ ∩

X

τ6=γ∈G

(KI)γ = (0)✳ ❆ss✐♠✱ ❛ s♦♠❛ X

τ∈G

(KI)τ é ❞✐r❡t❛✳ ❆❞❡♠❛✐s✱ é ❢á❝✐❧ ✈❡r q✉❡ KI =

M

τ∈G

(KI)τ✳ ❆❣♦r❛✱ ❞❛❞♦s mα ∈K, rβ ∈Iβ ❡ rτ ∈ Rτ✱ t❡♠♦s

(mαrβ)rτ =mα(rβrτ),

❥á q✉❡ mα ∈ M ❡ M é ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦✳ ❈♦♠♦ I é ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦✱ t❡♠♦s q✉❡ rβrτ ∈ Iβ+τ✳ ▲♦❣♦✱ KI é ✉♠ s✉❜♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❞❡

M✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♣❧✐❝❛♥❞♦ ♦ ❧❡♠❛ ❛♥t❡r✐♦r ♣❛r❛ K = {mα} ∈ Mα✱ ♦❜t❡♠♦s q✉❡mαI é ✉♠ R✲s✉❜♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ ❞❡ M✳

✶✳✸ ➪❧❣❡❜r❛s ●r❛❞✉❛❞❛s

◆❡st❛ s❡çã♦✱ r❡❝♦r❞❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ F✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s✱ ♦♥❞❡F é ✉♠ ❝♦r♣♦ ❡ Gé ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✜♥✐t♦✳ ❆s ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛q✉✐ sã♦ ❛♥á❧♦❣❛s ❛s ❛♣r❡s❡♥t❛❞❛s ♥❛ ❙❡çã♦ ✶✳✶✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ F✲á❧❣❡❜r❛ é ✉♠ ❛♥❡❧✳

❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ F✲á❧❣❡❜r❛ A é G✲❣r❛❞✉❛❞❛ s❡ A ♣♦❞❡ s❡r ❡s❝r✐t❛

❝♦♠♦ ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ F✲s✉❜❡s♣❛ç♦s

A=M

α∈G

Aα ✭✶✳✸✮

t❛✐s q✉❡ ♣❛r❛ q✉❛✐sq✉❡r α, β ∈G, AαAβ ⊆ Aα+β. ❉✐r❡♠♦s q✉❡ A é ✉♠❛ s✉♣❡rá❧❣❡❜r❛ s❡ G=Z2✳

❯♠❛ ✈❡③ q✉❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ t❡♠♦s ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✱ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡A♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ❢♦r♠❛ ú♥✐❝❛ ❝♦♠♦ s♦♠❛ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s✳

❯♠ s✉❜❡s♣❛ç♦ B ⊆ A é ❣r❛❞✉❛❞♦ ♦✉ ❤♦♠♦❣ê♥❡♦ s❡ B = M

α∈G

(B ∩ Aα). ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ B é ❣r❛❞✉❛❞♦ s❡✱ ♣❛r❛ q✉❛❧q✉❡r b = X

α∈G

bα ∈ B, t❡♠✲s❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ bα ∈ Bα ♣❛r❛ t♦❞♦ α ∈ G. ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r s✉❜á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❡ ✐❞❡❛✐s ❣r❛❞✉❛❞♦s✳

✶✷ ❉❡✜♥✐çã♦ ✶✳✸✳✷✳ ❙❡❥❛♠ A ❡ B ❞✉❛s F✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❤♦♠♦✲ ♠♦r✜s♠♦ ❞❡F✲á❧❣❡❜r❛sϕ :A −→ B é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡F✲á❧❣❡❜r❛sG✲❣r❛❞✉❛❞❛s ✭♦✉ ❤♦♠♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ❞❡ ❣r❛✉ ♥❡✉tr♦✮ s❡✱ ♣❛r❛ t♦❞♦ α ∈ G✱ t❡♠♦s ϕ(Aα)⊆ Bα✳ ❙❡ ϕ ❢♦r t❛♠❜é♠ ❜✐❥❡t✐✈♦✱ ❞✐③❡♠♦s q✉❡ ϕ é ✉♠ ✐s♦♠♦r✜s♠♦ ✭♦✉ ✐s♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ❞❡ ❣r❛✉ ♥❡✉tr♦✮ ❞❡ F✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s ❡ A ❡ B sã♦ F✲á❧❣❡❜r❛s G✲❣r❛❞✉❛❞❛s ✐s♦♠♦r❢❛s✳

❊♠ ❣❡r❛❧✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❤♦♠♦♠♦✜s♠♦s ❞❡ ❛♥é✐s ✭F✲á❧❣❡❜r❛s✮ ❣r❛❞✉❛❞♦s ❞❡ ❣r❛✉ ♥❡✉tr♦✳

❉❡✜♥✐çã♦ ✶✳✸✳✸✳ ❯♠❛ F✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛ A é ❞✐t❛ s✐♠♣❧❡s s❡ A2 _{6= (0) ❡ A ♥ã♦}

❝♦♥té♠ ✐❞❡❛✐s ❜✐❧❛t❡r❛✐sG✲❣r❛❞✉❛❞♦s ♥ã♦ tr✐✈✐❛✐s✳

❈♦♠♦ ✐❧✉str❛çã♦ ❞❛s ❞❡✜♥✐çõ❡s ❛❝✐♠❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ❞♦✐s ❡①❡♠♣❧♦s ❝❧áss✐❝♦s ❞❡ á❧✲ ❣❡❜r❛s ❣r❛❞✉❛❞❛s✳ ❊st❛s á❧❣❡❜r❛s sã♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❛s á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s s✐♠♣❧❡s✳

❊①❡♠♣❧♦ ✶✳✸✳✶✳ [ [✻], Example 2.1] ❙❡❥❛ R =Mn(F) ❛ á❧❣❡❜r❛ ❞❡ ♠❛tr✐③❡s n×n s♦❜r❡ ✉♠ ❝♦r♣♦ F ❡ s❡❥❛ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦✳ ❋✐①❡ ✉♠❛ n✲✉♣❧❛ α= (α1, . . . , αn) ∈ Gn ❞❡

❡❧❡♠❡♥t♦s ❞❡ G✳ ❊♥tã♦✱ ❛ n✲✉♣❧❛ α ❞❡✜♥❡ ✉♠❛ G✲❣r❛❞✉❛çã♦ ❡♠ R ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❙❡❥❛♠ eij,1 ≤ i, j ≤ n✱ ♠❛tr✐③❡s ✉♥✐tár✐❛s ❞❛ á❧❣❡❜r❛ R✳ P❛r❛ ❝❛❞❛ β ∈ G, ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦

Rβ =Span{eij | αj−αi =β}.

❱❡r✐✜❝❛✲s❡ ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡ RτRβ ⊆ Rτ+β ♣❛r❛ q✉❛✐sq✉❡r τ, β ∈G ❡✱ ♣♦rt❛♥t♦✱

R=M

τ∈G

Rβ

é ✉♠❛ G✲❣r❛❞✉❛çã♦✳ ❊st❛ G✲❣r❛❞✉❛çã♦ é ❝❤❛♠❛❞❛ ❞❡ ❣r❛❞✉❛çã♦ ❡❧❡♠❡♥t❛r ❞❡✜♥✐❞❛ ♣❡❧❛ n✲✉♣❧❛ α✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ [ [✻], Example 2.4] ❙❡❥❛ R = F[G] ❛ á❧❣❡❜r❛ ❞❡ ❣r✉♣♦ ❞❡ G s♦❜r❡ ✉♠ ❝♦r♣♦F✱ ♦✉ s❡❥❛✱ R=Span{rα |α ∈G}❝♦♠ ♦ ♣r♦❞✉t♦rαrβ =rα+β✱ ♦♥❞❡{rα | α∈G}é ✉♠❛ ❜❛s❡ ❞❡ R✳ ❆ á❧❣❡❜r❛ R é ❡q✉✐♣❛❞❛ ❝♦♠ ❛ G✲❣r❛❞✉❛çã♦ ❝❛♥ô♥✐❝❛R =M

α∈G

Rα✱ ♦♥❞❡

Rα = Span{rα} é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✶ ❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s ♥ã♦ ♥✉❧♦s sã♦ ✐♥✈❡rtí✈❡✐s✳

✶✸ ❉❡✜♥✐çã♦ ✶✳✸✳✹✳ ❯♠❛F✲á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛ é ❝❤❛♠❛❞❛ ❞❡ á❧❣❡❜r❛ G✲❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐✲ sã♦ s❡ é ✉♥✐tár✐❛ ❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❤♦♠♦❣ê♥❡♦ ♥ã♦ ♥✉❧♦ ♣♦ss✉✐ ✐♥✈❡rs♦✳

❖❜s❡r✈❛♠♦s q✉❡ s❡Dé ✉♠❛F✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐sã♦✱ ❡♥tã♦D0é ✉♠❛F✲á❧❣❡❜r❛

❞❡ ❞✐✈✐sã♦✳

❈❧❛r❛♠❡♥t❡✱ t♦❞❛ F✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐sã♦ é ✉♠❛ F✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ s✐♠♣❧❡s✳ P♦ré♠✱ ❛ r❡❝í♣r♦❝❛ ♥ã♦ é ✈❡r❞❛❞❡✐r❛✳ P♦r ❡①❡♠♣❧♦✱ ❛F✲á❧❣❡❜r❛ ❞❡ ♠❛tr✐③❡s ❝♦♠ ❣r❛❞✉❛çã♦ ❡❧❡♠❡♥t❛r é ❣r❛❞✉❛❞❛ s✐♠♣❧❡s✱ ♠❛s ♥ã♦ é ✉♠❛F✲á❧❣❡❜r❛ ❣r❛❞✉❛❞❛ ❞❡ ❞✐✈✐sã♦✳

❆t❡♥t❛♠♦s ❛✐♥❞❛ q✉❡ F✲á❧❣❡❜r❛s ❣r❛❞✉❛❞❛s ❞❡ ❞✐✈✐sã♦ ❡ ❛♥é✐s ❣r❛❞✉❛❞♦s ❞❡ ❞✐✈✐sã♦ ♥ã♦ ❝♦♥tê♠ ✐❞❡❛✐s ✉♥✐❧❛t❡r❛✐s ❣r❛❞✉❛❞♦s ♥ã♦ tr✐✈✐❛✐s✳

❉❡✜♥✐çã♦ ✶✳✸✳✺✳ ❯♠ ♠ó❞✉❧♦ à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✮ ❣r❛❞✉❛❞♦ s♦❜r❡ ✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ ❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦ é ❝❤❛♠❛❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✮ ❣r❛❞✉❛❞♦✳

❙❡❥❛♠D✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ ❣r❛❞✉❛❞♦ ❞❡ ❞✐✈✐sã♦ ❡M ✉♠D✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ à ❞✐r❡✐t❛✳

❯♠ ❝♦♥❥✉♥t♦ {m1, m2, . . . , mk} ⊆ h(M) éD✲❞❡♣❡♥❞❡♥t❡ ✭♦✉ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡ s♦✲

❜r❡D✮ s❡ ❡①✐st❡♠ ❡❧❡♠❡♥t♦sd1, . . . , dn∈h(D)✱ ♥ã♦ t♦❞♦s ♥✉❧♦s✱ t❛✐s q✉❡

k

X

i=1

midi = 0.P♦r ❬✶✵❪✱ ❬✷✹❪ ❡ ❬✷✽❪✱ t♦❞♦s ♦s r❡s✉❧t❛❞♦s ♣❛❞rõ❡s ❞❡ á❧❣❡❜r❛ ❧✐♥❡❛r✱ ❝♦♠♦ ✐♥❞❡♣❡♥❞ê♥❝✐❛ ❧✐♥❡❛r ❡ ❝♦♥❥✉♥t♦ ❣❡r❛❞♦r✱ ❝♦♥t✐♥✉❛♠ ✈❛❧❡♥❞♦ ♣❛r❛ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ ❛♥é✐s ✭F✲á❧❣❡❜r❛s✮ ❣r❛❞✉❛❞♦s ❞❡ ❞✐✈✐sã♦✱ ♣♦✐s sã♦ ♠ó❞✉❧♦s ❧✐✈r❡s✱ ✈❡❥❛ ❬❬✷✹❪✱ Pr♦♣♦s✐t✐♦♥ ✹✳✻✳✶❪ ❡ ❬❬✷✽❪✱ Pr♦✲ ♣♦s✐t✐♦♥ ✷✳✺❪✳ ▼♦str❛♠ t❛♠❜é♠ q✉❡ t♦❞❛s ❛s ❜❛s❡s ❤♦♠♦❣ê♥❡❛s ❞❡ M tê♠ ❛ ♠❡s♠❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❡✱ ♣♦rt❛♥t♦✱ ❢❛③ s❡♥t✐❞♦ ❢❛❧❛r ❞❡ ❞✐♠❡♥sã♦ ❞♦ ♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦ M✱ ❛ q✉❛❧ ❞❡♥♦t❛♠♦s ♣♦r dimD(M)✳ ❈♦♠♦ D0 é ✉♠ ❛♥❡❧ ✭F✲á❧❣❡❜r❛✮ ❞❡ ❞✐✈✐sã♦✱ t❡♠♦s

q✉❡ ❝❛❞❛ Mα✱ ♣❛r❛ t♦❞♦ α ∈ G✱ é ✉♠ D0✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ à ❞✐r❡✐t❛✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ ✉♠

❝♦♥❥✉♥t♦ {m1, m2, . . . , mk} ⊆ Mα é D✲❞❡♣❡♥❞❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ {m1, m2, . . . , mk} é

D0✲❞❡♣❡♥❞❡♥t❡✳ ◆ˇastˇas❡s❝✉ ❡ ❖②st❛❡②❡♥ ♠♦str❛r❛♠ ❡♠ ❬✷✹❪ q✉❡ s❡Dé ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦

❞❡ ❞✐✈✐sã♦✱ ❡♥tã♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✶✮ M é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡D❀

✷✮ M t❡♠ ❜❛s❡ ✜♥✐t❛ s♦❜r❡ D❀

✸✮ M t❡♠ ✉♠❛ ❜❛s❡ ❤♦♠♦❣ê♥❡❛ ✜♥✐t❛ s♦❜r❡ D✳

✶✳✹ ❍♦♠♦♠♦r✜s♠♦s ●r❛❞✉❛❞♦s

✶✹ ❙❡❥❛M ✉♠ R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛G✲❣r❛❞✉❛❞♦✳ ❉❡♥♦t❛♠♦s ♣♦rEnd(M)♦ ❛♥❡❧ ❞❡ t♦❞♦s

♦s ❡♥❞♦♠♦r✜s♠♦s ❞❡ M✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡End(M)é ✉♠ ❛♥❡❧ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s✱

s♦♠❛ ❡ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✳

❯♠ ❡♥❞♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ✭♦✉ ❤♦♠♦❣ê♥❡♦✮ ❞❡ ♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s ❞❡ ❣r❛✉ α é ✉♠ ❡♥❞♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦f :M −→M t❛❧ q✉❡

Mβf ⊆Mα+β

♣❛r❛ q✉❛❧q✉❡r β ∈G.

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡♥❞♦♠♦r✜s♠♦s ❣r❛❞✉❛❞♦s ❞❡ ❣r❛✉α ❝♦♥st✐t✉❡♠ ✉♠ s✉❜❣r✉♣♦ ❛❞✐t✐✈♦ End(M)α ❞♦ ❛♥❡❧ End(M)✳ ❈♦♥s✐❞❡r❡ Endgr(M) =

X

α∈G

End(M)α✳ ➱ r❡❧❛t✐✈❛✲

♠❡♥t❡ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡Endgr_{(M)é ✉♠ s✉❜❛♥❡❧ ❞❡ End(M)❡ q✉❡ ❛ s♦♠❛} X α∈G

End(M)α

é ❞✐r❡t❛✳ ▲♦❣♦✱ Endgr_{(M) =} M α∈G

End(M)α é ❛♥❡❧ ❣r❛❞✉❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ M é ✉♠

R✲♠ó❞✉❧♦ ✜❡❧✱ ❡♥tã♦ R ⊆Endgr_{(M)✈✐❛ ♦ ♠♦♥♦♠♦r✜s♠♦ ❞❡ ❛♥é✐s ❣r❛❞✉❛❞♦s} ϕ: R −→ Endgr_(M)

r 7−→ R_{r :M →M}

m7−→mr.

❙❡❥❛♠ M = M

α∈G

Mα ❡ N = M

β∈G

Nβ R✲♠ó❞✉❧♦s à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦s✳ ❉❡♥♦t❡♠♦s ♣♦r

HomR(M, N) ♦ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛❞✐t✐✈♦ ❞❡ t♦❞♦s ♦s ❤♦♠♦♠♦r✜s♠♦s ❞❡R✲♠ó❞✉❧♦s ❣r❛❞✉✲

❛❞♦s ❞❡M ❡♠ N✳

❯♠❛ ❛♣❧✐❝❛çã♦ R✲❧✐♥❡❛r f : M −→ N é ❝❤❛♠❛❞❛ ❞❡ ❤♦♠♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ✭♦✉ ❤♦♠♦❣ê♥❡♦✮ ❞❡ ❣r❛✉α✱α ∈G✱ s❡

Mβf ⊆Nα+β

♣❛r❛ q✉❛❧q✉❡r β ∈G.

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❤♦♠♦♠♦r✜s♠♦s ❣r❛❞✉❛❞♦s ❞❡ ❣r❛✉ α é ✉♠ s✉❜❣r✉♣♦ ❛❞✐t✐✈♦ HomR(M, N)α ❞♦ ❣r✉♣♦ HomR(M, N). ❈♦♥s✐❞❡r❛♠♦s

Homgr_R(M, N) = X

α∈G

HomR(M, N)α.

❋❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛✲s❡ q✉❡ ❛ s♦♠❛ é ❞✐r❡t❛ ❡✱ ♣♦rt❛♥t♦✱ Homgr_R(M, N) =

M

α∈G

HomR(M, N)α é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❣r❛❞✉❛❞♦✳ ❖❜s❡r✈❛♠♦s q✉❡ Hom gr

✶✺ ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ ❢✉♥çõ❡s✱ q✉❡ ❞❡♥♦t❛♠♦s ♣♦r Endgr_R(M)✳

❉✐③❡♠♦s q✉❡Endgr_R(M) é ♦ ❛♥❡❧ ❞❡ ❡♥❞♦♠♦r✜s♠♦s ❣r❛❞✉❛❞♦s ❞♦R✲♠ó❞✉❧♦ M✳

❉❡ ♠♦❞♦ ❣❡r❛❧✱ Homgr_R(M, N) ( HomR(M, N)✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ q✉❡ ❡①✐st❡ f ∈

HomR(M, N) ❡ f /∈ HomgrR(M, N)✱ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬ ❬✷✸❪✱ ❊①❛♠♣❧❡ ✷✳✹✳✶❪✳ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ♠♦str❛ ❡♠ q✉❡ ❝♦♥❞✐çõ❡s ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡✳

❈♦r♦❧ár✐♦ ✶✳✹✳✶✳ [ ❬✷✸❪✱ ❈♦r♦❧❧❛r✐❡s ✷✳✹✳✹✲✷✳✹✳✻] ❙❡ M é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡♥tã♦ Homgr_R(M, N) = HomR(M, N)✳

❖❜s❡r✈❡ q✉❡ s❡ ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ R é ✉♥✐tár✐♦✱ ❡♥tã♦ Endgr_R(RR) = EndR(RR) ≃ R sã♦ ✐s♦♠♦r❢♦s ❝♦♠♦ ❛♥é✐s ❣r❛❞✉❛❞♦s ✭✈❡❥❛ ❬✷✸❪✮✳

P❛r❛ ❝❛❞❛ ❤♦♠♦♠♦r✜s♠♦ ❤♦♠♦❣ê♥❡♦ f : M −→ N✱ ♦s s✉❜♠ó❞✉❧♦s Ker(f) ⊆ M ❡ Img(f) ⊆ N sã♦ t❛♠❜é♠ ❣r❛❞✉❛❞♦s✳ ❙❡ N ⊆ M sã♦ ♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s✱ ❡♥tã♦ ♦ ❡♣✐♠♦r✜s♠♦ ❝❛♥ô♥✐❝♦ π: M −→M/N é ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ 0✳ ❖❜✈✐❛♠❡♥t❡✱ ❛ ❛♣❧✐❝❛çã♦

✐❞❡♥t✐❞❛❞❡1 :M −→M é ❤♦♠♦❣ê♥❡❛ ❞❡ ❣r❛✉ 0✳

✶✳✺ ❆♥é✐s ●r❛❞✉❛❞♦s Pr✐♠✐t✐✈♦s

❉✐s❝✉t✐r❡♠♦s ❛q✉✐ r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❡str✉t✉r❛❧ ❞❡ ❛♥é✐s ❣r❛❞✉❛❞♦s ♣r✐♠✐t✐✈♦s✱ ♦s q✉❛✐s sã♦ ❛♥á❧♦❣♦s ❛♦s ❞❛ t❡♦r✐❛ ❞❡ ❛♥é✐s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬✶✾❪✱ ❬✷✷❪ ❡ ❬✷✺❪✳

❉❡✜♥✐çã♦ ✶✳✺✳✶✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦✳ ❉✐③❡♠♦s q✉❡ R é ✉♠ ❛♥❡❧ G✲❣r❛❞✉❛❞♦

♣r✐♠♦ s❡ ♣❛r❛ q✉❛✐sq✉❡r ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s G✲❣r❛❞✉❛❞♦s ♥ã♦ ♥✉❧♦s I ❡ J ♦❝♦rr❡ IJ 6= (0).

▼♦str❛r❡♠♦s q✉❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✐❞❡❛✐s ✉♥✐❧❛t❡r❛✐s ❣r❛❞✉❛❞♦s✳ ▲❡♠❛ ✶✳✺✳✶✳ ❯♠ ❛♥❡❧G✲❣r❛❞✉❛❞♦RéG✲❣r❛❞✉❛❞♦ ♣r✐♠♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ q✉❛✐sq✉❡r

✐❞❡❛✐s à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✮ ❣r❛❞✉❛❞♦s ♥ã♦ ♥✉❧♦s I ❡ J ❞❡ R t❡♠✲s❡ IJ 6= (0)✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛R ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ♣r✐♠♦✳ ❙❡❥❛♠ I ❡ J ✐❞❡✐❛s à ❞✐r❡✐t❛ ❣r❛❞✉❛❞♦s ♥ã♦ ♥✉❧♦s ❞❡R✳ ❊♥tã♦✱RI ❡RJ sã♦ ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s ❣r❛❞✉❛❞♦s ❞❡R✳ P❡❧♦ ▲❡♠❛ ✶✳✷✳✶✱

Annr

R(R) é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❣r❛❞✉❛❞♦ ❞❡ R✳ ❆❧é♠ ❞✐ss♦✱ (AnnrR(R))2 = (0). ❈♦♠♦

R é ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ♣r✐♠♦✱ t❡♠♦s q✉❡ Annr

R(R) = (0). ❆ss✐♠✱ RI ❡ RJ sã♦ ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s ❣r❛❞✉❛❞♦s ♥ã♦ ♥✉❧♦s ❞❡ R✳ ◆♦✈❛♠❡♥t❡✱ ❝♦♠♦ R é ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ♣r✐♠♦✱ RIRJ 6= (0)✳ ▲♦❣♦✱IRJ 6= (0) ❡ (0)6=IRJ ⊆IJ. P♦rt❛♥t♦✱ IJ 6= (0).