If R is a field (or more generally a skew-field) and S is not the zero ring, then f is injective. If both R and S are fields, then im(f) is a subfield of S, so S can be viewed as a field extension of R. If R and S are commutative and I is an ideal of S then f−1(I) is an ideal of R. If R and S are commutative and P is a prime ideal of S then f−1(P) is a prime ideal of R. If R and S are commutative, M is a maximal ideal of S, and f is surjective, then f−1(M) is a maximal ideal of R. If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R. If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R. If f is surjective, P is prime (maximal) ideal in R and , then f(P) is prime (maximal) ideal in S. Moreover, The composition of ring homomorphisms is a ring homomorphism. For each ring R, the identity map is a ring homomorphism. Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings. The zero map sending every element of R to 0 is a ring homomorphism only if S is the zero ring (the ring whose only element is zero). For every ring R, there is a unique ring homomorphism . This says that the ring of integers is an initial object in the category of rings. For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a terminal object in the category of rings. Examples The function , defined by is a surjective ring homomorphism with kernel nZ (see modular arithmetic). The complex conjugation is a ring homomorphism (this is an example of a ring automorphism). For a ring R of prime characteristic p, is a ring endomorphism called the Frobenius endomorphism. If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring. (Otherwise it fails to map 1R to 1S.) On the other hand, the zero function is always a rng homomorphism. If R[X] denotes the ring of