Ring homomorphism

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 ${\displaystyle f:R\to S}$ that preserves addition, multiplication and multiplicative identity; that is,^{[1][2][3][4][5]}

${\displaystyle {\begin{aligned}f(a+b)&=f(a)+f(b),\\f(ab)&=f(a)f(b),\\f(1_{R})&=1_{S},\end{aligned}}}$

for all ${\displaystyle a,b}$ in ${\displaystyle R.}$

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.

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