Question

6. Prove or disprove: If Ri and R, are commutative rings with unity, then Ri Re is a commutative ring with unity.

I need a full explanation on #6 please.

Answer 1

8) Given that R, and Ro be commitative ring with unity set e, and en be the identity or unity element of R, and R2 respectively Now, define R, x R 2 = {(a, b)|vaER, , &BER 2} and (a,b), (x,y) ER, XR2 then (a, b) ② (ny) = (atx, 6 ty) (a, b) (ny) = (ax, bu y) where Rut, i) and (Re, tzie) fare ring we show that (R, XR2, 6 , ) forms a ring. 1) Since CR., t,, :) and (R₂, te, 2) are ring then (R., +) and (R2, te) are commutative group. Then (Rex Re, @) is a commutative group. 2) let &=(a,b), B = (2, 4) E R XR 2 then a KER, and by the COB= (a, b) (2, 4) = (aux, 6;Y) ER, XR2 (Rit, i) is a ring an ER, ANER? Land (R2, +2; 2) is aring b, YER2 by ER2 KOBER, XR, V&BER, XR -

and bi ER, 14, 23 a-16.e) = (0.6).e Vi-,2 3 let &(a, b), B= (an, bz), 8= (an, bz) t R, XRq Then a; ER, Then &O (B08) = (a, b) ol cas, be) ① (03, 63)) = (a, b) (az az , beibg) = (a (ag-as) , b , ( 62 063)). In (R; , +,-) ring ab,etR;] - = 1 (2,-2). Az , (6, 62) - 63 ) albel-cable. = [ca;-e), (baj62) ) © (03, b) - ((a., 6,) 0 (02,43) (03,64) = (XOB) og &O (BBY)= (&BBOY & A, B, 8 € R, XR2 4) 20(B 8) = (0,by) (02,6») ® (ag, by)) = (1, 6.) 0 (aztaz, bat bg) = (ai (aztag), b, (bet 63)) [Ri,t,') ring a, b, c ER;) a. (b+ C) = arb ta.c &i=1,2) = (a; aq +0,as , bibat bibg) Lai = (a.az, b, bz) (a.az, b, bo). = ((9, 6) (22, 62)) (ca. 6.) 6 (an, bz)) = ( AB) (&or) for x, B, 8 E R XR 2

D), Similarly we can show that (COB)DY = (08) (303) W&B,Y ERAB 8 Now, a = (a, b, ) ER&Ra then . A,ER, and stRg and also loce, e) ER, & Rq where e; is the identity element a in Ri j = 1, 2 so nei = einx & HERi , ; 21,2 Then &o e = (a, b, ) o (e, , ez) = (a, e,, B, C) = (a, b)=4 . and eox=(e., eq) 6 (a, b) = (e. a, e, be) = (2,6)=x . : 20 e = e0 x = 4 V&ER, XR2 jlis the identity element in R, XR2 ::(R, XRq ,0 ,0) satisfying all the condition of Ring and e is the identity element of R, KR2 R, XR2 is a ring with identity, ut N = (a;, 6;) € R, XR₂ , 1=1,2 then , ON2 = (a,, 6.0 (A2, bz) = (a.az, b, bz) Ri is ] = (a2a, brab) = (a2, bz) 0 (9, 6) = x₂ on ring non2 = Xon, VN; ER, RR2, ;= 1, 2 Rix Ra is also commutative ring with limity. commutativel