1) Consider the mapping p: Z3 --> Z6 given by p(x) = 2x, for x = 0,1,2. Is p a ring homomorphism? How about the mapping p(x) = remainder of 4x (mod 6)?
You first need to consider if is well-defined. Say that then , so and hence which means the mapping is well-defined. Now what about ? Try to argue that this is also well-defined. To show it is a ring homorphism (depending on how you define "ring homomorphism") you just need to show and . Notice . Also, . Thus, it is not a homorphism, you try doing the second case.