Mathematics | Ring Homomorphisms

• Last Updated : 24 Nov, 2021

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]IR  IS  , if IR and IS 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(OR) = f(OS) where OR and OS 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

1.  , ∀a, b ∈ .
2. , ∀a, b ∈ .
3. IF  IK , where IF and IK are identities of set over and set  over  operations respectively.
4. OF  OK , where OF and OK are identities of set  over  and set  over  operations respectively.

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