Define f Z18 Z6 times Z3 by fa18 3a6 a3 where xn is the eq

Define f: Z_18 - Z_6 times Z_3 by f([a]_18) = ([3a]_6, [a]_3) where [x]_n is the equivalence class of x modulo n. (a) Is f is a homomorphism of rings? (Explain) (b) Is f an isomorphism of rings? (Explain)

Solution

With f given as above, clearly

f(x+y) = ((x+y) (mod 6),(x+y) (mod 3))

          = (x mod 6, x mod 3) + (y mod 6, y mod 3)

          = f(x) + f(y)

Similarly f(xy) = f(x)f*y)

So f is a homomorphism of rings

(b) f(6) =f912) =(0,0)

So f is not one -one .

So f is not an isomorphism