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Ring homomorphism

In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. See Also Ring theory.

If R and S are rings and f : R -> S is a function, we require

• f(a + b) = f(a) + f(b) for all a and b in R
• f(ab) = f(a) f(b) for all a and b in R
• f(1) = 1

Properties

Directly from these definitions, one can deduce:

• f(0) = 0
• f(-a) = -f(a)
• 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 R_p is the smallest subring contained in R and S_p is the smallest subring contained in S, then every ring homomorphism f : R -> S induces a ring homomorphism f_p : R_p -> S_p. 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.

Examples

• The function f : Z -> Z_n, defined by f(a) = [a]_n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• There is no ring homomorphism Z_n -> Z.
• If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, 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 M_n(R) -> M_n(S).