all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function defined by (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. The kernel of f consists of all polynomials in R[X] that are divisible by . If is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the matrix rings . Let V be a vector space over a field k. Then the map given by is a ring homomorphism. More generally, given an abelian group M, a module structure on M over a ring R is equivalent to giving a ring homomorphism . A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear. Non-examples The function defined by is a rng homomorphism (and rng endomorphism), with kernel 3Z/6Z and image 2Z/6Z (which is isomorphic to Z/3Z). There is no ring homomorphism for any . If R and S are rings, the inclusion sending each r to (r,0) is a rng homomorphism, but not a ring homomorphism (if S is not the zero ring), since it does not map the multiplicative identity 1 of R to the multiplicative identity (1,1) of . The category of rings Endomorphisms, isomorphisms, and automorphisms A ring endomorphism is a ring homomorphism from a ring to itself. A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven rngs of order 4. A ring automorphism is a ring isomorphism from a ring to itself. Monomorphisms and epimorphisms Injective ring homomorphisms are identical to monomorphisms in the category of rings: If is a monomorphism that is not injective, then it sends some