If f is a homomorphism, is it true that f-1(x+y)=f-1(x)+f-1(y) and f-1(xy) = f-1(x) * f-1(y)??
If a function is not bijective then the inverse is not defined.(There are things that will fail if a function is not bijective. for example having a left inverse is related to being one-to-one, having a right inverse is related to being surjective, if a function isn't a bijection then it won't have inverses(it can be proved that the inverse of a function is unique easily)) Hence, it doesn't make sense to talk about the inverse of a homomorphism if it's not defined. in fact homomorphisms that are bijective are special, they are called isomorphisms as you know.
If f is a ring homomorphism with kernel K, then the condition is equivalent to , and the condition implies that The first of those is true, the second one is not.
For example, if R is a ring, S is the ring R[x] of polynomials over R, and the homomorphism f:S→R is the map , then K is the ideal of polynomials with zero constant term. But consists of those elements of K for which the coefficient of x is also 0. Therefore