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

1.  , ∀a, b ∈ .
2.  , ∀a, b ∈ .
3. [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 :

1. Function f(x) = x mod(n) from group (,+,*) to (_n,+,*) ∀x ∈  is group of integers. + and * are simple addition and multiplication operations respectively.
2. Function f(x) = x for any two groups (R,+,*) and (S,⨁,) ∀x ∈ R, which is called identity ring homomorphism.
3. Function f(x) = 0 for groups (N,*,+) and (Z,*,+) for ∀x ∈ N.
4. 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 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

1.  , ∀a, b ∈ .
2. , ∀a, b ∈ .
3. I_F  I_K , where I_F and I_K are identities of set over and set  over  operations respectively.
4. O_F  O_K , where O_F and O_K are identities of set  over  and set  over  operations respectively.