Homomophism

RING HOMOMORPHISM

Definition 1 - 22
Let R and R/ be two rings . A ( ring ) homomorphism from R into R/
is meant a function ƒ : R ? R/ ?
ƒ ( a+ b ) = ƒ ( a ) + ƒ ( b ) , ƒ ( a.b ) = ƒ ( a ) . ƒ ( b )
for every a , b ? R .
A homomorphism which is a one – to – one mapping is called an
isomorphism .

Notes
The + and . on the left – hand sides of the equations in the definition
are of R , whereas the + and . on the right - sides are of R/ .

If ƒ is a homomorphism from R into R/ , then the image ƒ ( R) of R
under ƒ is called the homomorphic image of R . When R = R/ ƒ being
a homomorphism or R into itself is called endomorphism of R .
An isomorphism of R onto itself is called an automorphism of R .
Example 1
Let ƒ : R ? R/ , where ƒ (a) = 0 ? a ? R . Prove that ƒ is homomorphism.
Solution
Let a , b ? R , then
ƒ ( a + b ) = 0 = 0 + 0 = ƒ ( a ) + ƒ (b )
ƒ ( a . b ) = 0 = 0 . 0 = ƒ ( a ) . ƒ ( b ) .
So that , ƒ is homomorphism.
This mapping is called trivial homomorphism
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