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Mathematics | Ring Homomorphisms

  • Last Updated : 24 Nov, 2021

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

  1. f (a + b) = f (a) ⨁ f (b)   , ∀a, b ∈ R  .
  2. f(a * b) = f(a) \times  f(b)   , ∀a, b ∈ R  .
  3. f  [Tex](  [/Tex]IR)   =   IS  , if IR and IS 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 :

  1. 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.
  2. Function f(x) = x for any two groups (R,+,*) and (S,⨁,\times  ) ∀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,⨁,\times   ) 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,⨁,\times  ) then f : (R,+) → (S,⨁) is group homomorphism.

Ring Isomorphism :
A one 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

  1. f(a + b) = f(a) ⨁ f(b)   , ∀a, b ∈ F  .
  2. f(a * b) = f(a)  \times  f(b)  , ∀a, b ∈ F  .
  3. f(  IF)   =   IK , where IF and IK are identities of set F   over *   and set K   over \times   operations respectively.
  4. f(  OF)   =   OK , where OF and OK are identities of set F   over +   and set K   over ⨁   operations respectively.

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