Ring homomorphism

In ring theory a ring homomorphism is a map from one ring to another group that preserves the ring operations.

Formally, therefore, a map $f:A\rightarrow B$ is a homomorphism if

$f:0_{A}\mapsto 0_{B};\,$

$f:-x\mapsto -f(x);\,$

$f:x+y\mapsto f(x)+f(y);\,$

$f:x.y\mapsto f(x).f(y).\,$

A unital homomorphism between unital rings (rings with a multiplicative identity element) must also satisfy

$f:1_{A}\mapsto 1_{B}.\,$

The kernel of a homomorphism is the set of all elements of the domain that map to the zero element of the codomain. This subset is an ideal, and every ideal is the kernel of some homomorphism.

An embedding or monomorphism is an injective homomorphism (or, equivalently, one whose kernel consists only of the zero element).

Examples

• The zero map is a ring homomorphism.
• The map defined on the ring of integers which maps an integer to its remainder modulo N for some fixed modulus N is a ring homomorphism from Z onto Z/(N).

Isomorphism

We say that two rings are isomorphic if there is a bijective homomorphism of one onto the other : the mapping is called an isomorphism. Isomorphic rings have identical structure and are often thought of as just being relabelings of one another.

References

• A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 34-37. ISBN 0-521-09695-2.