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Let
be ring. A map is a ring homomorphism if , the following hold -
The Kernel
is the set defined as Where
is the additive element of -
A Ring isomorphism
is a one-to-one and onto homomorphism. We say that rings are isomorphic denoted -
Ring homomorphisms are a generalization of group homomorphisms. All results concerning group homomorphisms are valid for the additive structure of rings
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(Fraleigh 26.3) Ring homomorphisms preserve subring structures (i.ee., subrings correspond to subrings under homomorphisms) in the same way that group homomorphisms preserve subgroup structure.
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(Fraleigh 26.5) If
and , then where is the Coset containing of the commutative additive group -
(Fraleigh 26.6)
is one-to-one if and only if -
(Fraleigh e26.21) If
be a ring homomorphism such that , If has unity and has no divisors, then is unity for
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(Fraleigh 26.7) Let
be a ring homomorphism with . Then the additive cosets of form a ring whose binary operations are defined by choosing representatives. That is Also the map
defined by Is an isomorphism.
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(Fraleigh 26.16) The Fundamental Homomorphism Theorem The Fundamental Homomorphism Theorem. Every Factor Ring gives rise to a natural homomorphism.
Let
be a homomorphism with . Then
is a ring and given by is an isomorphism. In other words If
is onto, then the isomorphism changes to If
is the homomorphism given by , then We refer to
and as the natural isomorphism and natural homomorphism respectively.
Special Examples
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Let
be the ring of all functions mapping . The evaluation homomorphism is defined for each . as where For
and . -
Let
be rings. For each , the map defined by is the projection homomorphism
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The Frobenius homomorphism is a map
for a Commutative Ring with unity where for prime . It is defined as