Aein Inenionl Jounl of Conepoy eeh Vol 4 No ; Jnuy 204 Pil nd Wekly Pil ub ei odule lik Bineh ub l hei Depen Jodn Univeiy of iene nd Tehnology Ibid 220 Jodn Ab Le be ouive eiing wih ideniy nd n -ei odule The onep of pil ub odule h been inodued nd udied []Alo wekly pil ub odule hve been udied [2] Thoughou hi wok, we define pil nd wekly pil ub ei odule new genelizion of pie ub ei odule We how h hey enjoy of ny of he popeie of pie ub ei odule hei ubje Clifiion: 3C05 Key Wod: ubeiodule, Pil nd Wekly Pil ubeiodule Inoduion Thoughou will be ouive eiing wih nonzeo ideniy nd i n -eiodule Pil nd wekly pil idel ove ouive eiing hve been udied [4] The onep of pie ubeiodule h been inodued by G Yeilo, K Oel nd U Teki [3] Thoughou hi wokwe inveige viou popeie of pil nd wekly pil ubeiodulend hei genelizion We lo exploe he elionhip beween pilnd wekly pil ubeiodule well he elionhip beween weklypil ubeiodule ove nd wekly pil ubeioduleove,whee i he e of ll nelble eleen in oeove we define P-pil ubeiodule nd udy i elion wih pie ubeiodule ofeiodule nd ohe eiodule uue Finlly we eblih oneo-one oepondene beween P- wekly pil ubeiodule of n -eiodule nd he P -wekly pil ubeioduleof 2 Pil nd Wekly Pil ubeiodule G Yeilo,K Oelnd U Teki[3], defined h ube N of n -eiodule i lled ubeiodule of if fo n,n N nd, n+n N nd n N,n eleen i id o be pie o N if N(wih ) iplie h N,h i, ( N : ) =N nd N i pie ub-eiodule of he -eiodule if foeh nd, N iplie ( N : )o N We will denoe he e of ll eleen of h e no pie o N by (N) Definiion 2 A pope ubeiodule N of i lled pil ubeiodule of if (N) fo n idel of Le N pil ubeiodule nd P=(N),hen we y h N i P pil ubeiodule of Theoe22 Le be ouive eiing nd be n -eiodule If N i P pil ubeiodule of,hen P i pie idel Poofuppoe h, \P We will how h P If hee i uh h ( N, hen () N, o ( N : ) =N ine i pie o N, N iplie ( N : )=N Thi how N, h i i pie o N, o P equied Popoiion23 Le N be ubeiodule of n -eiodule If N i P-pil ubeiodule of,hen ( N : ) P Poof Le ( N : ) nd 0(ine0 P)ine N i pope ubeiodule of, ( N : ) i pope idel of, hee exi \N, wih N Thi how h i no pie o N, h i P 3
Cene fo Pooing Ide, UA wwwijneo Popoiion24Le be ouive eiing, n -eiodule N nd K -ubeiodule of wih K N nd P n idel of Then he following hold (i) I f K i P pil ubeiodule of N nd (N) P, hen K i P pil ubeiodule of (ii) Le (,P) be lol eiing If K i P pil ubeiodule of N nd N i pil ubeiodule of, hen K i pil ubeiodule of Poof (i) If i no pie o K,hen hee exi \K wih K If N,hen P o uppoe h N Theefoe, (N) p Now le P, we will how h h i no pie o K Thee exi n N\K uh h n K Thu n \K give i no pie o K Thu K i P pil ubeiodule of (ii)thi follow fo (i) Le be given eiing, be he e of ll ulipliively nelble eleen of nd ben - eiodule Then he eiodulegeneed by in, i he e of ll finie u +22++nn, whee i nd i,whih i denoed by If N i ubeioduleof, define N ={ ; N} Clely N i ubeiodule of G Yeilo, K Oelnd U Teki [3], defined h he k-loue of ub-eiodule N of -eiodule i +=2, fo oe i N, i=,2} nd ubeiodule N i lled k-ubeiodule if N equl N={ : i k-loue Le25 Aue h i eiing Le be -eiodule nd N be ubeiodule of he eiodule Then he following hold: (i) X if nd only If i n be wien in he fo (ii) (N ) = N x fo oe nd (iii) If N i pie ubeiodule of hen N i pie ubeiodule of - Poof (i) Le x Then hee e eleen nd i uh h x = i,wihi = i,heefoe x= i n i= ii whee i nd i Pu =2n fo uible eleen e,e2,,en, we hve i = ei The Convee Ipliion i Obviou e nen,whee ei (ii)(n ) N i ivil Fo he evee inluion, ifx N, hen x = nen nd, o =x N nd odingly, x=, whee =e + 2e2 + + (iii)aue h N i pie ubeiodule of nd N, N i pie ubeiodule,hen ( N : ) o N, h i ( N : N ) o N Theoe26Le be n -eiodule nd he e of ulipliively nelble ube of If K i Q pil ubeiodule of,hen K i Q pil ubeiodule of 32
Aein Inenionl Jounl of Conepoy eeh Vol 4 No ; Jnuy 204 Poof Le N=K nd P=Q I i enough o how h (N)=PFi,le (N) Thee exi \N uh h N Then fo K nd \ K we ge Q,hene p On he ohe hnd,fo evey p,we hve Q, o hee exi K uh h K Then by le 25 (ii), K, o K wih k \ K N wih \ ( K ) Thi how h i no pie o N, h i (N) Definiion 27 Le be ouive eiing nd be n -ubieodule A pope ubeiodule N of i lled wekly pie ubeiodule if wheneve 0 N, fo oe nd, hen eihe N o N Le be ouive eiing nd N be ubeiodule of n - eiodule An eleen i lled wekly pie o N if 0 N( ) iplie h N Ohewie i no wekly pie o NDenoe By w(n) he e of eleen of h e no wekly pie o N Le N be ubeiodule of he -eiodule Then 0 i lwy wekly pie o N nd if i pieo N,hen i wekly pie o N, bu he onve i no neeily ue Definiion28 Le be ouive eiing wih non -zeo ideniy nd le N be popeubeiodule of n -eiodule N i lled wekly pil if he e P=w(N) {0} fo n idel of P i lled he (wekly) djoin idel of N nd we lo y N i P wekly pil ubeiodule of We nex give evel heizion of wekly pil ubeiodule Theoe29Le P be n idel of ouive eiing, n - eiodule nd N ubeiodule of Then he following eequivlen : (i) N i P wekly pil (ii) Fo evey P {0}, ( N : N (0 : ; nd fo evey 0 P, N ( 0 : ( N : Poof ( i) ( ii) uppoe h N i P wekly pil ubeiodule of Le P {0} = w(n) nd le ( N : If = 0 hen ( 0 : If 0, ine i wekly pie o N, we ge N o N ( 0 :, ine N ( 0 : N fo ny ubeiodule, we hve ( N : N (0 : Now uppoe h P {0} = w(n), i no wekly pie o N, ohee exi \N uh h 0 N, ( N : \ ( N(0 : ) ( ii) ( i) I follow fo (ii) h w(n) = P {0} Hene N i P wekly pil Theoe 20 Le be ouive eiing, be n -eiodule, nd N i ubeiodule of if N i P wekly pil ubeiodule of hen P i wekly pie idel of Poof uppoe h, b P wih b 0 we will how h b P If hee i wih 0 (b) N, hen 0 ( N : b) N (0 : b) by Theoe 29, whee ( 0 : b), 0 A P, hen i wekly pie o N, hene N h i b i wekly pie o N, b P Conequenly P i wekly pie idel of Le be ouive eiing An -eiodule i lled fihful(ep, yli) if Ann() =0, h i ( 0 : ) 0(ep, n be geneed by ingle eleen ie; = (x) = x) Popoiion 2 Le N be fihful k-ubeiodule of n -eiodule If N i P wekly pil ubeiodule of, hen ( N : ) P Poof Aue h N i P wekly pil ine N i pope ubeiodule of N, hen ( N : ) i pope idel of, hee exi \N Le non- zeo eleen ( N : ), hen N If 0 nd N give h i no wekly pie o N, h i P 33
Cene fo Pooing Ide, UA wwwijneo o we ue h = 0 N fihful ubeiodule hee exi n N wih n 0Now (n + ) N wih + n N ine N i k-ubeiodule hi iplie h i no weklypie o N, h i P Thu ( N : ) P equied Theoe 22 Le be ouive eiing, N i k-ubeiodule of fihful yli -eiodule, If N i P wekly pil of hen ( N : ) i P wekly pil idel of PoofLe = x fo oe x, nd e I = ( N : ) We will how h w(i) = w(n) Fo evey w(i), hee exi \I uh h 0 I, x 0 ohewie ( 0 : x) (0 : ) 0, ine i fihful x \N, i follow h i no wekly pie o N, h i w(n) Now ue h w(n), hen 0 N fo oe \N, we n wie = x fo oe, 0 x N, 0 I wih \I, hi how i no wekly pie o I Hene w(i),w(i) = w(n) equied We nex give ondiion fo wekly pil ubeiodule o be pil ubeiodule Theoe 23 Le be ouive eiing nd P i k-idel of If k-ubeiodule N of - eiodule i P wekly pil of wih ( N : ) P nd N ( N : ) 0 Then N i pil ubeiodule of PoofI i enough o how h P = (N) Fo evey 0 P, i no wekly pie o N, o i no pie o N, h i (N) Now ue h (N) Thee exi \N uh h N If 0 hen i no wekly pie o N h i P o ue h = 0 Ce()uppoe h N 0, y n 0 0 fo oe n0 N Now 0 ( + n 0 ) Nwih + n \N, iplie h i no wekly pie o N, nd hene P e(2) uppoe h N = 0 If ( N : ) 0, hen 0 fo oe ( N : ) Now 0 ( + N wih \N iplie h + i no wekly pie o N, h i + P nd hene P ine P i k-idelif ( N : ) = 0 ine N ( N : ) 0, hee i (N : ) nd n N wih n 0 Now 0 (+)(+n) N wih +n \N how h + P wih P iplie P, ine P i k-idel Theefoe (N) Pnd o (N) = P whih iplie h N i P-pil Theoe 24 Le be ouive eiing nd be n -eiodule Then fihful k wekly pie ubeiodule N of i wekly pil Poof Le N be k wekly pie ubeiodule of wih Ann ( N) 0 e P = ( N : ) N pope ubeiodule o h ( N : ) fo evey P\{0},hee exi \N uh h N If 0, hen w(n) o we n ue = 0 N i fihful ubeiodule, hee exi n N wih n 0, 0 (n + ) N wih n + \N, iplie h i no wekly pie, h i w(n) Now ue h w(n), hee exi \N wih 0 N A N i wekly pie we ge ( N : ) \{0}, hene w(n) P\{0} We hve ledy how h P = w(n) {0} Hene N i P wekly pil Theoe 25 Le be ouive eiing, be he e of ulipliively nelbleelen of, nd N k- ubeiodule of If N i P wekly pil of wih P = Then he following hold (i) If 0 / N, hen N (ii) ( N : ) = N : ) ( Poof (i) uppoe h 0 / N nd \N, hen hee exi n N nd wih wih N, hene i no weklypie o N whih i ondiion 0 (ii) Le ( N : ) fo evey, N, by (i) N n, 0 = n 34
Aein Inenionl Jounl of Conepoy eeh Vol 4 No ; Jnuy 204 Thi iplie h ( N : ), h i ( N : ) nd hene ( N : ) ( N : ) The ohe oninen i obviou Popoiion 26 Le be ouive eiing, he e of ulipliively nelble eleen of nd N be k ubeiodule of eiodule If N i P wekly pil wih P hen he following hold (i) N i P wekly pil ubeiodule of -eiodule (ii) N = N Poof (i)le 0 P hen by Theoe 20, P i wekly pie idel nd 0 P by [4, le 8] Thee exi \N uh h 0 N, we u hve 0 N, by popoiion 426 N, hen i no wekly pie o N, o P \{0} w(n) Now uppoe h w( N ), hen 0 N fo oe \ N, hen 0 N, by Popoiion 25, 0 N wih \N, hi how P\{0}, hen P \{0} (ii) Le N If = 0 hen N If 0, hen 0 N by Popoiion 25 N The ohe oninen i obviou Theoe 27 Le be n -eiodule nd he e of ulipliively nelble eleen of If K i Q wekly pil ubeioduleof, henk i (Q ) wekly pil ubeiodule of Poof e N = K nd P = Q By [4, popiion 30], P i wekly pie idel of I i enough o 0 how h w(n) = P\{0} Fo evey w(n) hee exi \N uh h 0 N, Knd \K h Q\{0} o P\{0} Thi how w(n) P\{0} Now le P\{0}, hen Q\{0} K i Q wekly pil ubeiodule of 0 wih K, hen 0 equied, hee exi \K K,0 N wih \N Thu w(n), h i P\{0} w(n) Theoe 28 Le P be wekly pie idel of ouive eiing nd le be n -eiodule Aue h i ulipliively nelble ube of wih P = Then hee exi one-o-one oepodene beweenhe P-wekly pil ubeiodule of nd he P wekly pil ubeiodule of Poof Thi follow fo Popoiion 26 nd Theoe 27 efeene J Dun, Pil odule, Co Algeb 997; 25: 2409-2435 EbhiiAni nd A YouefinDni, Wekly pil ubodule, Tkng Jounl of hei 2009; 40: 239-245 G Yeilo, K Oel nd U Teki On pie ubeiodule Inenionl Jounl of Algeb (200); 4: 53-60 EbhiiAni,On Pil nd wekly pil idel ove ouive eiing, Gl 2008; 43: 3-23 35