for(a), (r,0) has no inverse if r is not 0!

for(b), It is a field, because the mapping: (a,b)--> a+ib, (i is the imaginary unit) is a isomorphism. Or you can varify that all axioms for field are satisfied.

for(c), (a) can't be a field for the same reason.

(b) can't be a field because the inverse of some element is not unique.