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(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 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.
^Artin 1991, p. 353
^Eisenbud 1995, p. 12
^Jacobson 1985, p. 103
^Lang 2002, p. 88
^Hazewinkel 2004, p. 3
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