Examples of Ring Homomorphism
- Function f(x) = x mod(n) from the group ([Tex]Z [/Tex],+,*) to ([Tex]Z [/Tex]n,+,*) ∀x ∈ [Tex]Z, Z [/Tex] is a group of integers. + and * are simple addition and multiplication operations respectively.
- Function f(x) = x for any two groups (R,+,*) and (S,⨁,[Tex]\times [/Tex]) ∀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 a complex conjugate form group (C,+,*) to itself, here C is a set of complex numbers. + and * are simple addition and multiplication operations respectively.
NOTE: If f is a homomorphism from (R,+,*) and (S,⨁,[Tex]\times [/Tex] ) 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 a ring homomorphism from (R,+,*) and (S,⨁,[Tex]\times [/Tex]) then f : (R,+) → (S,⨁) is a group homomorphism.
Ring Isomorphism:
A and onto homomorphism from ring [Tex]R [/Tex] to ring [Tex]S [/Tex] is called Ring Isomorphism, and [Tex]R [/Tex]and [Tex]S [/Tex]are Isomorphic.
Ring Automorphism:
A homomorphism from a ring to itself is called Ring Automorphism.
Field Homomorphism:
For two fields [Tex](F,+,*) [/Tex] and [Tex](K,⨁, \times) [/Tex] a mapping [Tex]f : F → K [/Tex] is called field homomorphism if
- [Tex]f(a + b) = f(a) ⨁ f(b) [/Tex] , ∀a, b ∈ [Tex]F [/Tex].
- [Tex]f(a * b) = f(a) \times f(b) [/Tex], ∀a, b ∈ [Tex]F [/Tex].
- [Tex]f( [/Tex]IF[Tex]) [/Tex] [Tex]= [/Tex] IK , where IF and IK are identities of set [Tex]F [/Tex]over [Tex]* [/Tex]and set [Tex]K [/Tex] over [Tex]\times [/Tex] operations respectively.
- [Tex]f( [/Tex]OF[Tex]) [/Tex] [Tex]= [/Tex] OK , where OF and OK are identities of set [Tex]F [/Tex] over [Tex]+ [/Tex] and set [Tex]K [/Tex] over [Tex]⨁ [/Tex] operations respectively.
Mathematics | Ring Homomorphisms
Ring Homomorphisms are a concept from abstract algebra that plays a crucial role in various applications, such as cryptography, coding theory, and systems theory. Understanding ring homomorphisms helps in the study and application of algebraic structures and their properties.