Question

4. Prove that isomorphism is an equivalence relation on the set of all rings. You may assume elementary results regarding the inverses and compositions of bijections. Proof:

Answer 1

! TA is a râng ? rang AnB iff there rexists an isomer chism between A and B ! Reflexive. fet AES Id: A -> A deftinedd a'sin!! Illa)= a for all att Them Id is the frivial isomorphism ANA for all Aest fymmetric Let AnB their there resist an is omrophism tos y P1A 77>Bi fi & is one. Que onto so 8: B A resist and is one one on to to $1$'(*) + plius) = 1 ple)) + + ($lus) - Q+ yet $(2) + $!9). 17; (***) $1$les. Fla)), = + (4582). oploca)) { $, is somethiom) -'; (23. fly) = "cak You mi¢!: B~> A to an isomolehians is Brea

Transitive fet AnB and Br G then there exists two issomorphism off d bw All, A LA DE & Jof: A G is an ibomorphism m g * of are bijeehive - Joof so bilechve! 18of)(a+b) = (f14+b)} = f(flas + flbs) if is an isomorphism = (fear) +g(feb02 g is an isomorphismi = Ljóf)la9 + fofjl5). (gof)(4.b) = f (f Carb)) -(flas. feb . fiban isemorphism = g (feas . g (tebi n g is an Isomorphism = (40f) (a), 1 gof)(b) - Jof: AV G is an isomorphesing An G Thus the relation me. is reflexive, Symmetic & framsitive trewe it is an reque Valence relation,