Ring homomorphism

In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity; that is,
for all a, b in R.
These conditions imply that additive inverses and the additive identity are also preserved.
If, in addition, is a bijection, then its inverse ⁻¹ is also a ring homomorphism. In this case, is called a ring isomorphism, and the rings R and S are said to be isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
If R and S are [rng (algebra)|]s, then the corresponding notion is that of a homomorphism, defined as above except without the third condition f = 1_S. A homomorphism between rings need not be a ring homomorphism.
The composition of two ring homomorphisms is a ring homomorphism. It follows that the rings form a category with ring homomorphisms as morphisms.
In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

Properties

Let be a ring homomorphism. Then, directly from these definitions, one can deduce:

f = 0_S.
f = −f for all a in R.
For any unit a in R, f is a unit element such that. In particular, f induces a group homomorphism from the group of units of R to the group of units of S.
The image of f, denoted im, is a subring of S.
The kernel of f, defined as, is a two-sided ideal in R. Every two-sided ideal in a ring R is the kernel of some ring homomorphism.
A homomorphism is injective if and only if its kernel is the zero ideal.
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 homomorphism exists.
If R_p is the smallest subring contained in R and S_p is the smallest subring contained in S, then every ring homomorphism induces a ring homomorphism.
If R is a division ring and S is not the zero ring, then is injective.
If both R and S are fields, then im is a subfield of S, so S can be viewed as a field extension of R.
If I is an ideal of S then ⁻¹ is an ideal of R.
If R and S are commutative and P is a prime ideal of S then ⁻¹ is a prime ideal of R.
If R and S are commutative, M is a maximal ideal of S, and is surjective, then ⁻¹ is a maximal ideal of R.
If R and S are commutative and S is an integral domain, then ker is a prime ideal of R.
If R and S are commutative, S is a field, and is surjective, then ker is a maximal ideal of R.
If is surjective, P is prime ideal in R and, then f is prime ideal in S.

Moreover,

The composition of ring homomorphisms and is a ring homomorphism.
For each ring R, the identity map is a ring homomorphism.
Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
The zero map that sends every element of R to 0 is a ring homomorphism only if S is the zero ring.
For every ring R, there is a unique ring homomorphism. This says that the ring of integers is an initial object in the category of rings.
For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a terminal object in the category of rings.
As the initial object is not isomorphic to the terminal object, there is no zero object in the category of rings; in particular, the zero ring is not a zero object in the category of rings.

Examples

The function, defined by is a surjective ring homomorphism with kernel nZ.
The complex conjugation is a ring homomorphism.
For a ring R of prime characteristic p, is a ring endomorphism called the Frobenius endomorphism.
If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the zero ring . On the other hand, the zero function is always a homomorphism.
If R 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 defined by is a surjective ring homomorphism. The kernel of f consists of all polynomials in R 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 not a ring homomorphism, but is a homomorphism, with kernel 3Z/6Z and image 2Z/6Z.
There is no ring homomorphism for any.
If R and S are rings, the inclusion that sends each r to is a rng homomorphism, but not a ring homomorphism, since it does not map the multiplicative identity 1 of R to the multiplicative identity of.

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. On the other hand, up to isomorphism, there are eleven [Rng (algebra)|]s 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 r₁ and r₂ to the same element of S. Consider the two maps g₁ and g₂ from Z to R that map x to r₁ and r₂, respectively; and are identical, but since is a monomorphism this is impossible.
However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion with the identity mapping is a ring epimorphism, but not a surjection. However, every ring epimorphism is also a strong epimorphism, the converse being true in every category.