"Let R be the ring Zp[x] of polynomials with coefficients in the finite field Zp, and let f:R->S be a surjective homomorphism from R to a ring S. Show that S is either isomorphic to R, or is a finite ring." If S is isomorphic to R, then we're done. If S is not isomorphic to R, then by hypothesis, f is not injective. That is, ker(f) contains more than the zero element of R. ker(f) is an ideal of R, therefore there exists a surjective homomorphism p:R->R/ker(f), where p(x) = x + ker(f). My question is, if R is homomorphic to S and R is homomorphic to T, what is the relation between S and T? Are they the same? Isomorphic? In the question, R and S are given. In my answer, S is constructed.