Definition (Definition 22.9) Let R and R' be rings. A map o : R - R' is a homomorphism if the following two conditions are satisfied for all a, b E R; pla + b) = p(a) + o(b) and 2 p( ab) = p(a) $(b).Definition (Definition 22.12) An isomorphism o : R - R' from a ring R to a ring R' is a ring homomorphism that is one-to-one and onto R'. The rings R and R' are isomorphic if there exists an ring isomorphism from R to R'. Example (Example 22.12) As abelian groups, it is easy to show that (Z, +) and (27, +) are isomorphic under the map o : Z -> 2Z with o(x) = 2x for x E Z. However, o is not a ring isomorphism because p(xy) = 2xy but &(x) $(y) = (2x) (2y) = 4xy.as groups 42. Show that the rings 27) and 3Z)are not isomorphic. Show that the fields R and C are not isomorphic. as rings. Let $:27 -37 be a ring isomorphism with 8 (2 ) = Q E 3Z. Since $ 14 ) = $ ( 2+ 2) = $( 2) + $(2) = ata = 20. and $ 14 ) = $ (2 .2) = $ ( 2 ) . $ (2 ) = a. a = a 9 a2 = 20 . This implies a = 0 or a= 2 . Prove that it's impossible!42. Show that the rings 2Z and 3Z are not isomorphic. Show that the fields R and C are not isomorphic. Consider an isomorphism 4 : 4 - R. Then $ ( 1) = 1 . and $ ( 1 ) = - 1 Show that it's impossible using P ( i)