A NOTE ON PRE-J-GROUP RINGS

Transcription

1 Qaar Univ. Sci. J. (1991), 11: A NOTE ON PRE-J-GROUP RINGS By W. B. VASANTHA Deparmen of Mahemaic, Indian Iniue of Technology Madra , India. ABSTRACT Given an aociaive ring R aifying amb = Bam for any pair a and b of i elemen, i being a poiive ineger, and a group G aifying xm- 1 = ym- 1, we give a ufficien and neceary condiion on he group ring GR o ha i elemen aify he ame condiion aigned o R. A imilar reul can be obained if he group G i replaced by a emigroup. In hi noe we obain a neceary and ufficien condiion for a group ring RG o be a pre J-ring. We call group ring which are pre-j-ring a pre-j-group ring. The auhor in [1] call an aociaive and commuaive ring R o be a pre-j-ring if here exi a poiive ineger m uch ha xmy = xym for every x and y E R. Theorem 1. Le G be a commuaive group in which x- 1 = ym--- 1 for every x and y in G and R a pre-j-ring wih xmy = xym for every x,y in R, where m i an ineger> 0 Then he group ring RG i a pre-j-ring if and only if for every a and 13 wih formal repreenaion a I aigi and 13 = I l3i hi in RG we have aml3 - ( I ar gr) 13 = al3m - a ( I 137 h7 ). Proof. Given R i a pre-j-ring in which xym = xmy for every x andy in Rand m a poiive ineger and G i commuaive group in which x- 1 = y- 1 for every x,y in G. Given RG i a pre-j-group ring o prove ( I aigi)m I 13ihi - I aimgr ( I l3ihi) = I aigi ( I 13jhj)m I aigi ( I 13h) hold in RG for every a, 13 in RG where a= I aigi and 13 = I l3ihi wih l3i, ai E R

2 Pre-J-Group Ring and gi> hi E G for j = 1, 2,..., and i = 1, 2,...,. Since aml3 = al3m.we have ( I aigi)m I.13ihi = I aigi ( I l3ihi)m expanding boh ide of he above equaion we ge ( I ag) I l3ihi + (I (erm of he form ai.ai... ak,gi.gi... gk) j:::il x I aihi = ( 2: <Xigi) (I 13h)+,(.I aigi) x (I (erm of he form l3i l3k, i=l i '=l..., 13b hj.hk... hj). Now ince R i a pre-j-ring we have xmy = xym for all x,y in R and ince G i commuaive wih gm- 1 = hm- 1 we have hgm- 1 g = hmg = gmh. Hence ( I ag)( I 13ihi )-(I aigi)( I 13h). 1=1 Thu he ideniy (.I aigi)m'(~ 13jhj) ( I ag)( I 13h). l=l ={ I aigi)( I 13ihi)m - ( I <Xigi ) I 13h i='l il=l hold in RG for every a, 13 in RG. Converely uppoe in RG we have he ideniy (,I aigi)m I 13ihi - ( I ag ) ( I l3ihi) i=j i=~ = (I aigi) ( I l3ihi)m - ( I aigi )( I '13h), i=.i To how ha RG i a pre-j-ring. i.e. o how aml3 = al3m for every a, 13 in RG. Since R i a pre-j-ring and G i a commuaive group uch ha xm- 1 = ym- 1 for every x, y in G we have al3i = ail3 for all ai, 13i in R. 28

3 W. B. VASANTHA Le a =I aigi and 13 =.I l3ihi (ai> l3i E R and gi> hi E G). Uing he given Jl.=/1 ideniy.. aml3 = ~ L ag) 13 = al3m i o prove RG i a pre-j-group ring i i ufficien if we prove. ( L aimg) 13 = a ( L 13h). if! Conider he lef hand ide ( L ag) 13 L ~mg (13Jhl h) i=.i i=!i L al31gh L al3gh = L aif3imgihlm I (li~mgihm ~=J = I aigi (13Jmh mhm) i'=i = a L 13h( which i nohing bu he righ hand ide of he above equaliy. Hence we have aml3 = al3m for all a, 13 in RG. Remark 2. If S i a commuaive emi-group in which xmy = xym for every x and y in S and R a pre J-ring, hen he emi-group ring RS i a pre J-emi-group ring if and only if i aifie he following ideniy for every a and 13 in RS aml3 ~ ( L. aim) 13 = al3m - a ( L -13h) 29

4 Pre-J-Group Ring where o: = I o:ii and~ =.I ~;h; wih o:b ~i in Rand h;, i ins fori= 1, 2,..., J=l and j = 1, 2,...,. Proof a in he cae of Theorem 1. REFERENCE [1] Lab Jiang, n he rucure of pre-j-ring. Hung-Chong Chow 65h anniverary, volume pp , Mah. Re. Cener, Na. Taiwan Univeriy, Taipei. 30

5 ~~ ~~.. i i '1 m "Z 'I I ;; -~ ~ ':-.I R <... ~ ~ ~..,...,,.,.o.j~ J "-:-! """:~ ~~ cuu.=.. '-'"""""' 4:-JI~! bjoj G ~J ~ R ~ b J a ~~ i.?"j amp = bam : i)l,ji._il<:.ll L. LAS: RG b II Ub. J uji...o 1 ~ ~ (])k 1'-l..bLU dj.j. ~ ',r r.r-:..>-"..>' ~ ~ 1.?..>J ~!J. S bjoj ~ G b.>"jji..:.j~i l.j! ~L.!..o 31