# Ring homomorphism

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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*R*arises from some ring homomorphism in this way.*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*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*. This can sometimes be used to show that between certain rings_{p}*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. - 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 ker(*f*) is not trivial). - 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*) = [*a*]_{n}=*a***mod***n*is a surjective ring homomorphism with kernel*n***Z**(see modular arithmetic). - There is no ring homomorphism
**Z**_{n}→**Z**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*X*^{2}+ 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

- An injective ring homomorphism is called
*ring monomorphism*. - A surjective ring homomorphism is called
*ring epimorphism*. - A bijective ring homomorphism is called
*ring isomorphism*.