Question

R= (×:×) € I, -5 <×< 2)

Answer 1

Take (x+x)2=(2x)2=4x2=4x. From the other hand, you have (x+x)2=x+x=2x. So 4x=2x, i.e. 2x=0 for any x. In other words, x=−x for any x.

Take x+y=(x+y)2. Then you have x+y=x2+y2+xy+yx=x+y+xy+yx. From this you can conclude that xy+yx=0, i.e. xy=−yx=yx. The last equality follows from the first paragraph

Answer 2

Answer --

Define a relation on RX R = {(p.a) |P,9 € R} as follows: (a,b), (c,d) € RxR, (a, b) (c,d) a.d=b.c. For any [(a,b)]: [(c,d)] ER:= (RX R)/ ~, define the additions +(1), +(2), and the multiplication as follows: [(a, b)] +() [(c,d)] := [(a.d+b.c,b. d)], [(a, b)] +(2) [(c,d)] := [(a + c, b + d)], [(a,b)]. [(c,d)] := [(a.cb.d)]. (a) Show that (R, +(1), ) is a ring. (b) Show that (R, +(2), ) is a ring. (c) Are they