Mathematics | Ring Homomorphisms

Last Updated : 02 Jan, 2023

Prerequisite : Rings

Ring Homomorphism :
A set $R$ with any two binary operations on set $R$ let denoted by $+$ and $*$ is called ring denoted as $(R, +, *)$ , if $(R, +)$ is abelian group, and $(R, *)$ is semigroup, which also follow right and left distributive laws.

for two rings $(R,+,*)$ and $(S,⨁,$ [Tex]\times [/Tex] $)$ a mapping $f : R → S$ is called ring homomorphism if

$f (a + b) = f (a) ⨁ f (b)$ , ∀a, b ∈ $R$ .
$f(a * b) = f(a) \times f(b)$ , ∀a, b ∈ $R$ .
$f$ [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 $R$ over $*$ and set $S$ over $\times$ operations respectively.

NOTE : Ring $(S,⨁, \times )$ is called homomorphic image of ring $(R,+,*)$ .

Examples :

Function f(x) = x mod(n) from group ( $Z$ ,+,*) to ( $Z$ _n,+,*) ∀x ∈ $Z, Z$ is group of integers. + and * are simple addition and multiplication operations respectively.
Function f(x) = x for any two groups (R,+,*) and (S,⨁, $\times$ ) ∀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,⨁, $\times$ ) 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,⨁, $\times$ ) then f : (R,+) → (S,⨁) is group homomorphism.

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

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

Field Homomorphism :
For two fields $(F,+,*)$ and $(K,⨁, \times)$ a mapping $f : F → K$ is called field homomorphism if

$f(a + b) = f(a) ⨁ f(b)$ , ∀a, b ∈ $F$ .
$f(a * b) = f(a) \times f(b)$ , ∀a, b ∈ $F$ .
$f($ I_F $)$ $=$ I_K , where I_F and I_K are identities of set $F$ over $*$ and set $K$ over $\times$ operations respectively.
$f($ O_F $)$ $=$ O_K , where O_F and O_K are identities of set $F$ over $+$ and set $K$ over $⨁$ operations respectively.