(a) What does the question mean by 'determining the kernel'? I know that kernels are ideals and ideals are kernels. Since is a ring homomorphism from , then is an ideal of R.
Now, a subring I is an ideal of R if and for . (The subring I is closed under multiplication with arbitrary elements of R). Is this all I need to say?!
(b) I have no idea how to do this one...
(c) I think is a prime ideal of R, if it is a proper ideal of R such that and imply or . What else do I need to state?