# Mathematics | Ring Homomorphisms

**Prerequisite : **Rings

**Ring Homomorphism :**

A set with any two binary operations on set let denoted by and is called ring denoted as , if is abelian group, and is semigroup, which also follow right and left distributive laws.

for two rings and [Tex]\times [/Tex] a mapping is called ring homomorphism if

- , ∀a, b ∈ .
- , ∀a, b ∈ .
- [Tex]( [/Tex]I
_{R}I_{S }, if I_{R}and I_{S}are identities (if they exist which in case of Ring with unity) of set over and set over operations respectively.

NOTE :* Ring ** is called homomorphic image of ring **.*

**Examples :**

- Function f(x) = x mod(n) from group (,+,*) to (
_{n},+,*) ∀x ∈ is group of integers. + and * are simple addition and multiplication operations respectively. - Function f(x) = x for any two groups (R,+,*) and (S,⨁,) ∀x ∈ R, which is called identity ring homomorphism.
- Function f(x) = 0 for groups (N,*,+) and (Z,*,+) for ∀x ∈ N.
- Function f(x) = which is complex conjugate form group (C,+,*) to itself, here C is set of complex numbers. + and * are simple addition and multiplication operations respectively.

**NOTE :** *If f is homomorphism from (R,+,*) and (S,⨁,** ) then f(O _{R}) = f(O_{S}) where O_{R} and O_{S} are identities of set R over + and set S over ⨁ operations respectively.*

**NOTE :*** If f is ring homomorphism from (R,+,*) and (S,⨁,**) then f : (R,+) → (S,⨁) is group homomorphism.*

**Ring Isomorphism :**

A one one and onto homomorphism from ring to ring is called Ring Isomorphism, and and are Isomorphic.

**Ring Automorphism :**

A homomorphism from a ring to itself is called Ring Automorphism.

**Field Homomorphism :**

For two fields and a mapping is called field homomorphism if

- , ∀a, b ∈ .
- , ∀a, b ∈ .
- I
_{F}I_{K}, where I_{F}and I_{K}are identities of set over and set over operations respectively. - O
_{F}O_{K}, where O_{F}and O_{K}are identities of set over and set over operations respectively.