| 1. | Every ideal is the kernel of a ring homomorphism and vice versa.
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| 2. | We therefore have short exact sequences split by a ring homomorphism
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| 3. | Let \ phi : R \ to S be a ring homomorphism.
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| 4. | The mapping is a ring homomorphism from the ring of limited hyperreals to.
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| 5. | Then one checks that these two functions are in fact both ring homomorphisms.
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| 6. | This can sometimes be used to exclude the possibility of certain ring homomorphisms.
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| 7. | The ring homomorphism ? appearing in the above is often called a structure map.
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| 8. | Any bijective ring homomorphism is a ring isomorphism.
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| 9. | So,, and are commutative rings with identity and and are ring homomorphisms.
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| 10. | This is a consequence of the fact that ring homomorphisms must preserve the identity.
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