Question

Give an example with justification of a function f:R1-->R2,where(R1,+,.) and (R2,+,.) are rings and such that f:(R1,+)-->(R2,+) is a group homomorphism but f is not a ring homomorphism.?

Answer 1

`f: ZZ_3 -> ZZ_3` given by `f(x) = hat(2)*x`

Explanation:

Consider `f: ZZ_3 -> ZZ_3` given by `f(x) = hat(2)*x`

We find:

`f(hat(0)) = hat(2) * hat(0) = hat(0)`

`f(a+b) = hat(2) * (a+b) = hat(2) * a+hat(2) *b = f(a)+f(b)`

but:

`f(hat(1) * hat(1)) = hat(2) * (hat(1) * hat(1)) = hat(2) != hat(1) = (hat(2) * hat(1)) * (hat(2) * hat(1)) = f(hat(1)) * f(hat(1))`