Ring homomorphism
In ring theory, a branch of abstract algebra, 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 f : R → S such that f is:
- addition preserving:
- f(a + b) = f(a) + f(b) for all a and b in R,
- multiplication preserving:
- f(ab) = f(a)f(b) for all a and b in R,
- and unit (multiplicative identity) preserving:
- f(1R) = 1S.
| Algebraic structure → Ring theory Ring theory |
|---|
|
Rings
Related structures
|
|
Commutative rings
p-adic number theory and decimals
|
![{\displaystyle \mathbb {Z} [1/p]}](../I/ab7b4e5c999e46c7614c3264b1d80708f4363433.svg)
Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above.
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 cannot be distinguished.
If R and S are rngs, then the corresponding notion is that of a rng homomorphism, defined as above except without the third condition f(1R) = 1S. 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 class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.