In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. If R and S are rings and f : R->S is a function, we require
If a has a multiplicative inverse in R, then f(a) has a multiplicative inverse in S and we have f(a-1) = (f(a))-1. Therefore, f induces a group homomorphism from the group of units of R to the group of units of S.
The kernel of f, defined as ker(f) = {a in R : f(a) = 0} is an ideal in R. Every ideal in R arises from some ring homomorphism in this way. f is injective if and only if the ker(f) = {0}.
If f is bijective, then its inverse f -1 is also a ring homomorphism. f is called an isomorphism in this case, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R->S induces a ring homomorphism fp : Rp->Sp. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms R->S can exist.
The composition of two ring homomorphisms is a ring homomorphism; the class of all rings together with the ring homomorphisms forms a category.
The function f : Z->Zn, defined by f(a) = [a]n = amodn is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
There is no ring homomorphism Zn->Z.
If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbersR, and C denotes the complex numbers, then the function f : R[X] ->C defined by f(p) = p(i) (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] which are divisible by X2 - 1.
If f : R->S is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings Mn(R) -> Mn(S).
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