Algebraic structure → Ring theory Ring theory |
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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,^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}

for all in

These conditions imply that additive inverses and the additive identity are preserved too.

If in addition *f* is a bijection, then its inverse *f*^{−1} is also a ring homomorphism. In this case, *f* is called a **ring isomorphism**, and the rings *R* and *S* are called *isomorphic*. From the standpoint of ring theory, isomorphic rings have exactly the same properties.

If *R* and *S* are rngs, then the corresponding notion is that of a **rng homomorphism**,^{[a]} defined as above except without the third condition *f*(1_{R}) = 1_{S}. A rng homomorphism between (unital) rings need not be a ring homomorphism.

The composition of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a category with ring homomorphisms as morphisms (see Category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.

Let *f* : *R* → *S* be a ring homomorphism. Then, directly from these definitions, one can deduce:

*f*(0_{R}) = 0_{S}.*f*(−*a*) = −*f*(*a*) for all*a*in*R*.- For any unit
*a*in*R*,*f*(*a*) is a unit element such that*f*(*a*)^{−1}=*f*(*a*^{−1}) . In particular,*f*induces a group homomorphism from the (multiplicative) group of units of*R*to the (multiplicative) group of units of*S*(or of im(*f*)). - The image of
*f*, denoted im(*f*), is a subring of*S*. - The kernel of
*f*, defined as ker(*f*) = {*a*in*R*|*f*(*a*) = 0_{S}}, is a two-sided ideal in*R*. Every two-sided ideal in a ring*R*is the kernel of some ring homomorphism. - An homomorphism is injective if and only if 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*R*→*S*exists. - If
*R*is the smallest subring contained in_{p}*R*and*S*is the smallest subring contained in_{p}*S*, then every ring homomorphism*f*:*R*→*S*induces a ring homomorphism*f*:_{p}*R*→_{p}*S*._{p} - If
*R*is a field (or more generally a skew-field) and*S*is not the zero ring, then*f*is injective. - If both
*R*and*S*are fields, then im(*f*) is a subfield of*S*, so*S*can be viewed as a field extension of*R*. - If
*I*is an ideal of*S*then*f*^{−1}(*I*) is an ideal of*R*. - If
*R*and*S*are commutative and*P*is a prime ideal of*S*then*f*^{−1}(*P*) is a prime ideal of*R*. - If
*R*and*S*are commutative,*M*is a maximal ideal of*S*, and*f*is surjective, then*f*^{−1}(*M*) is a maximal ideal of*R*. - If
*R*and*S*are commutative and*S*is an integral domain, 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*. - If
*f*is surjective,*P*is prime (maximal) ideal in*R*and ker(*f*) ⊆*P*, then*f*(*P*) is prime (maximal) ideal in*S*.

Moreover,

- The composition of ring homomorphisms
*S*→*T*and*R*→*S*is a ring homomorphism*R*→*T*. - For each ring
*R*, the identity map*R*→*R*is a ring homomorphism. - Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
- The zero map
*R*→*S*that sends every element of*R*to 0 is a ring homomorphism only if*S*is the zero ring (the ring whose only element is zero). - 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. - 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.

- The function
*f*:**Z**→**Z**/*n***Z**, defined by*f*(*a*) = [*a*]_{n}=*a*mod*n*is a surjective ring homomorphism with kernel*n***Z**(see modular arithmetic). - The complex conjugation
**C**→**C**is a ring homomorphism (this is an example of a ring automorphism). - For a ring
*R*of prime characteristic*p*,*R*→*R*,*x*→*x*^{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 (otherwise it fails to map 1_{R}to 1_{S}). On the other hand, the zero function is always a rng homomorphism. - 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*] that are divisible by*X*^{2}+ 1. - If
*f*:*R*→*S*is a ring homomorphism between the rings*R*and*S*, then*f*induces a ring homomorphism between the matrix rings M_{n}(*R*) → M_{n}(*S*). - Let
*V*be a vector space over a field*k*. Then the map*ρ*:*k*→ End(*V*) given by*ρ*(*a*)*v*=*av*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*R*→ End(*M*). - A unital algebra homomorphism between unital associative algebras over a commutative ring
*R*is a ring homomorphism that is also*R*-linear.

- The function
*f*:**Z**/6**Z**→**Z**/6**Z**defined by*f*([*a*]_{6}) = [4*a*]_{6}is a rng homomorphism (and rng endomorphism), with kernel 3**Z**/6**Z**and image 2**Z**/6**Z**(which is isomorphic to**Z**/3**Z**). - There is no ring homomorphism
**Z**/*n***Z**→**Z**for any*n*≥ 1. - If
*R*and*S*are rings, the inclusion*R*→*R*×*S*that sends each*r*to (*r*,0) is a rng homomorphism, but not a ring homomorphism (if*S*is not the zero ring), since it does not map the multiplicative identity 1 of*R*to the multiplicative identity (1,1) of*R*×*S*.

Main article: Category of rings |

- 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. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven rngs of order 4. - A
**ring automorphism**is a ring isomorphism from a ring to itself.

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If *f* : *R* → *S* is a monomorphism that is not injective, then it sends some *r*_{1} and *r*_{2} to the same element of *S*. Consider the two maps *g*_{1} and *g*_{2} from **Z**[*x*] to *R* that map *x* to *r*_{1} and *r*_{2}, respectively; *f* ∘ *g*_{1} and *f* ∘ *g*_{2} 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.