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

More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that

• 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

(If one does not require rings to have a multiplicative identity then the last condition is dropped.)

The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms.

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 ∈ R : f(a) = 0} is an ideal in R. Every ideal in a commutative ring R arises from some ring homomorphism in this way, but this is never true for a non-commutative ring. f is injective if and only if the ker(f) = {0}. Note that in general, for rings with identity the kernel of a ring homomorphism is not a subring since it will not contain the multiplicative identity.
• The image of f, im(f), is a subring of S.
• 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.
• If R is a field, then f is either injective or f is the zero function. (Note, however, that if f preserves the multiplicative identity, then it cannot be the zero function.)
• If both R and S are fields, then im(f) is a subfield of S (if f is not the zero function).
• If R and S are commutative and S has no zero divisors, then ker(f) is a prime ideal of R.
• For every ring R, there is a unique ring homomorphism Z → R. This says that the ring of integers is an initial object in the category of rings.

Examples

• The function f : Z → Z_n, defined by f(a) = ^*_n = a mod n is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• There is no ring homomorphism Z_n → Z for n > 1.
• If Rdenotes 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^* 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).

Types of ring homomorphisms

• A bijective ring homomorphism is called ring isomorphism.
• A ring homomorphism whose domain is the same as its range is called a ring endomorphism.

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f:R→S is a monomorphism which is not injective, then it sends some r₁ and r₂ to the same element of S. Consider the two maps g₁ and g₂ from Z^* to R which map x to r₁ and r₂, respectively; f o g₁ and f o g₂ are identical, but since f is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

See also

• homomorphism
• ring theory

Ring theory

Homomorfismo de anillos | Omomorfismo di anelli | 环的同态与同构