ON THE EXTENSION OF WEAK ARMENDARIZ RINGS RELATIVE TO A MONOID

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1 wwweo/voue/vo9iue/ijas_9 9f ON THE EXTENSION OF WEAK AENDAIZ INGS ELATIVE TO A ONOID Eye A & Ayou Eoy Dee of e Nowe No Uvey Lzou 77 C Dee of e Uvey of Kou Ou Su E-: ABSTACT Fo oo we oue e oe of we - -Aez w e oo eezo of - -Aez we -Aez vee oee oeove e ove we - -Aez f oy f fo y e -y- ue u x T ove we - -Aez If e e I eue / I we - - Aez e we - -Aez wee y oy oee oo Ao we ow f fy oo P we - -Aez e we - N -Aez wee N uue ou oo Keywo: uue ou oo -Aez --Aez we --Aez AS Sue Cfo: 6S6 6U 6N6 6P6 6U99 INTODUCTION Touou e eoe ove o eey w ey oo eevey Gve e oyo ove eoe y [x] We eoe y T e -y ue u x ove Te uy of Aez w e y Aez [5] ee Cw [9] A e Aez f weeve oyo f x x x x x [ x] fy f e fo e Te ovee wy ue Soe oee of Aez ve ee ue ee Cw [9] Aeo Co [] K Lee [] Ho e [] Lee Wo [] Suy [5] oue e oo of - Aez A e -Aez f weeve oyo f x x x x x x [ ] fy x f e fo e Zou [7] ue eezo of Aez w e e -Aez wee oo A e - Aez f weeve eee [ ] fy e fo e wee A e eue f o ozeo oe eee eue e Aez y Aez [5 Le ] u of Aez e o Aez A e e f evey eoe e Aez e e y K Lee [] Su of -Aez e o -Aez y Zou [7] Su of -Aez e o -Aez y Suy [5] Lu Zo [8] Ioue e oo of we Aez A e we Aez f weeve oyo f x x [ ] fy x f e fo e Tey owe fo eouve e I u / I we Aez e we Aez we Aez f oy f fo y ue u x ove we Aez Wu Hu-fe [4] Ioue e oo of e we -Aez A e we -Aez f weeve oyo f x x x [ ] fy f x e fo e Eoy e [] Ioue e oo of - -Aez wee 9

2 A & Eoy O e Exeo of We Aez e oo A e - -Aez f weeve eee [ ] fy e fo e wee I [6] A e we -Aez f weeve eee [ ] fy e fo e Cey -Aez e we -Aez e oo e u-oo uue ou oo f fo y wo o-ey fe ue A B ee ex eee uuey eee e fo wee A B Te of u-oo ue e o ee Beee P [6] P [4] Fo exe ue e o ef oee oo uoo of fee ou oo-fee oe ou Evey u-oo o uy eee of fe oe ove y eu Eoy e [] Suy [5] Zou [7] ee Cw [9] Zou Zo [8] K Lee [] Wu Hu-fe [4] Cu Jo [6] we w vee oo eezo of -Aez - -Aez w we e we - -Aez We -Aez eve o oo Fo o e w wy fo e ey of If e [ ] ove eoe e oo Defo Le e oo A e we - -Aez f weeve eee [ ] fy e fo e wee Pooo Evey u of we - -Aez we - -Aez Poof I ovou We oue e foow oo ee [5] Coo P: Fo f e Le [ Pooo ] If eue e fe e oo P u e ovee o ue Le 4 [6 Le ] Aue x x o x x e u -oo Te eve e fo x f Teoe 5 Le e u -oo e of Te we - -Aez Poof Le [ ] fy Se e of e / eue By Le / fe e oo o - -Aez y [ Teoe 6] Ao e So e foow e - -Aez Tu fo e Pooo 6 Le e u -oo eue Te we - -Aez Poof Se eue ee e of P fo e eu Tu e eu foow fo Teoe 5

3 A & Eoy O e Exeo of We Aez Cooy 7 Le e u -oo fy e oo P Te we - - Aez Poof Le e u -oo fy e oo P Te y [ Teoe 6] - -Aez Tu we - -Aez e oee oo If fo y < e < < Le e y oee oo e Cooy 8 Le e y oy oee oo fy e oo P Te we - -Aez Cooy 9 If fe oo P e we - Z -Aez fo y x x x x x x x x x [ x x ] f e fo I [ Teoe 9] w ow f I eue e of u / I - -Aez e - -Aez Hee we ve e foow eu fo we - -Aez oey Teoe Le e y oy oee oo I e of If I eue we - -Aez e we - -Aez Poof Le [ ] e u [ ] w / I We we < < < < < < < < < We w ue fe uo o e y oy oee e Noe o ow fo y / I[ ] Tu we I fo w e / I we - - ve Aez If ee ex u e If < e < oo u Sy Hee Now uoe w u fo y f < w e We w ow fo y w w Te X fe e We we X { u} Se X { w} u We I f f < e w < < < u < < u u < < < w

4 A & Eoy O e Exeo of We Aez oo If e fo Tu < foow w oo Sy we ve e Fo y < w uo yoe we ve u y Te we ve I e I I I I eue Tu fo y W e Now fo foow Se u X u I u I I eue we ve Tu u o fo e - e we o e ove Cou oe we ove fo y w w Tu we - -Aez I uy y e e wy fo u Tu Teefoe y fe uo fo y e oo e oo-fee f e foow oey o: f e e u Cooy Le e ouve eve oo-fee oo If oe of e foow oo o e we - -Aez fe oo P / I we - -Aez fo oe e I of I eue Poof If ouve eve oo-fee e y eo [] ee ex oe o oe o Now e eu foow fo Cooy 8 Teoe Pooo Suoe we - -Aez If [ ] e u e Poof I foow ey fo e efo wee oeffe of I [6] Cu Jo owe we -Aez f oy f T we - Aez Fo we - -Aez e foow we w ve oe eu: Teoe Le e oo w Te we - -Aez f oy f fo y T we - -Aez

5 A & Eoy O e Exeo of We Aez Poof We oe y u of we - -Aez we - -Aez Tu f T we - -Aez e we - -Aez Coveey Le ] [ we we B B A A ] [ C C e eee of ] [ T Aue I ey o ee ee ex oo of ] [ ] [ T T efe y: Aue B A C

6 A & Eoy O e Exeo of We Aez 4 Te I foow Se we - -Aez ee ex u fo y y Le } { x Te

7 A & Eoy O e Exeo of We Aez 5 C A B Tu C B A o T B C A T ow T we - -Aez Cooy 4 Le e oo If - -Aez e fo y T we - -Aez Pooo 5 Le e oo A we - -Aez f oy f e v exeo T we - -Aez Poof I foow fo Teoe Fo Teoe oe y ue f we - -Aez e evey -y- fu x ove we - -Aez wee Bu e foow exe ee e oy Exe 6 Le e e F S Le e e e eee ] [ S wee e Te Bu o oe Tu S o we - -Aez Cey evey - -Aez we - -Aez u e ovee o ue y e foow exe Exe 7 Le e we - -Aez Te e

8 A & Eoy O e Exeo of We Aez o - -Aez y [ Exe 4] we 4 we - -Aez y u Teoe e y u of we - -Aez we - -Aez Pooo 8 Te of we - -Aez oe ue fe e ou Poof Le e e fe e ou of wee { } we - - Aez Suoe fo oe eee [ ] wee e eee of e ou Se [ ] Se e o o Tu [ ] Se we -- Aez we ve Now fo e ee ex ove ee u e If we e x{ : } e e Teefoe T e we - -Aez e eee u of eu f u e fo Sy ef eu eee e efe A eee eu f o ef eu ee o zeo vo Teoe 9 Le e e uve oo o of e eu eee Te we - -Aez f oy f o we - -Aez Poof Le e we - -Aez S Pu S[ ] We y ue w w e fo S[ ] Now we ve v w wee u v u We w ow 6

9 A & Eoy O e Exeo of We Aez Hee u v w uvw [ ] Se we - -Aez fo o u v w uvw S[ ] fo Tu S we - - Aez Te ovee foow fo Pooo Pooo Le e oo e e eoe of If e e e e foow ee e euve: we - -Aez; e e e we - -Aez Poof ovou e e e e u of Le [ ] e u Le e e e e e e Te Se e e e we - -Aez ee ex e e e e e u e e e e e Tu e e Le x{ } Te e e Tu T e we - -Aez Pooo Le e oo e I e of u Aez If I e we - -Aez Poof Le / I we - - [ ] e u Te Se / I we - -Aez we ve / I Hee I Se I e T e we - -Aez I [7 ooo ] w ow f eue -Aez e [ ] N - Aez wee oo N u -oo Eoy e [ ooo ] w ow f fe e oo P - -Aez e [ ] - N -Aez wee N 7

10 A & Eoy O e Exeo of We Aez u -oo Fo we - -Aez we ve e foow eu Pooo Le e oo N u -oo If eouve w o we - -Aez e [ ] we - N -Aez Poof Suoe [ ] u Te fo e fe oo P we - - Aez Tu fo oo P Now e eu foow fo Teoe 5 Hee T ow [ ] fe Le [6 Le ] Le e eouve oo If [ ] e Pooo 4 Le e oo N u -oo If eouve w o we - -Aez e [N] we - -Aez Poof I ey o ee ee ex oo of [ N][ ] [ ][ N] efe y Now uoe [ N] e u wee fo We w ow [ N] wee N fo Tu [ ][ N] we ve Aue Te By Pooo [ ] we - N -Aez fo Se we - -Aez fo Hee Le e [N] we - -Aez Teoe 5 Le e oo N u -oo If eouve w o we - -Aez e we - N -Aez Poof Suoe { [ N] By Wou o of eey we ue } { } w we Fo y eoe A { } Te [ ][ N] Noe fo y A A w Now ey o ee ee ex oo of [ N] [ ][ N] efe y 8

11 A & Eoy O e Exeo of We Aez Suoe [ N] A Te fo e ove oo foow A B By Pooo [ ] we - N -Aez Tu we ve fo y C [ ] A B C Se we - -Aez A B y fo y C Tu fo T e we - N -Aez e oo Deoe { I I ee ex oy fe u e e Te oo w e oeo I I I Le I ey of } Cooy 6 Le I I e u -oo eouve If we - I Aez fo oe I e we - -Aez I Poof Le [ ] - u [ ] I [ ] fo oe fe ue } { } [ I -Aez Te { I Tu ] Te y Teoe 5 y uo we - -Aez o fo Hee we - Cooy 7 Le e oo eouve If we - -Aez e [x] [ x x ] e we - -Aez Poof Noe [ x] [ {}] [ x x ] [ ] ACKNOWLEDGENTS We wou e o e ee of Uvey of Kou Nowe No Uvey Ao e uo e efeee fo vey efu e of e e 9

12 A & Eoy O e Exeo of We Aez EFEENCES [] A Eoy E A L Zou O exeo of Aez eve o oo I J Pue A 9 No [] CY Ho NK K TK Kw O Sew Aez Co Ae No - [] DD Aeo V Co Aez Gu Co Ae 6 No [4] DS P Te Ae Suue of Gou Jo Wey New Yo 977 [5] EP Aez A oe o exeo of Be - J Au So [6] GF Beee JK P Tu x eeeo of exeo J Ae [7] L Zou Aez eve o oo Co Ae No [8] L Zou Zo O we Aez Co Ae [9] B ee S Cw Aez Po J A Se A S [] NK K Y Lee Aez eue J Ae [] P eo Noee of eeze owe ee J Pue A Ae [] TK Lee TL Wo O Aez Houo J 9 No [] Wu Hu-fe Exeo of eue J Hzou No U No [4] Wu Hu-fe O We -Aez J Hzou No U No 4-44 [5] Y Suy O e exeo of Aez [D] Lzou Uvey 8 [6] Z Cu C Jo we -Aez J Soue Uvey 5 No

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