Definitions

# Ring homomorphism

In ring theory or 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 : RS 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

Naturally, 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 (cf. the category of rings).

## 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 a commutative ring R arises from some ring homomorphism in this way. For rings with identity the kernel of a ring homomorphism is a subring without identity.
• The homomorphism f is injective if and only if the ker(f) = {0}.
• 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 there exists a ring homomorphism f : RS then the characteristic of S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphisms RS can exist.
• If Rp is the smallest subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : RS induces a ring homomorphism fp : RpSp.
• 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.
• If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R.
• For every ring R, there is a unique ring homomorphism ZR. This says that the ring of integers is an initial object in the category of rings.

## Examples

• The function f : ZZn, 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 ZnZ for n > 1.
• 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 : RS 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).

## Types of ring homomorphisms

• A bijective ring homomorphism is called a 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:RS 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[x] 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 ZQ is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.