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ring homomorphism
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(Definition)
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Let $R$ and $S$ be rings. A ring homomorphism is a function $f: R \longrightarrow S$ such that:
- $f(a+b) = f(a)+f(b)$ for all $a,b \in R$
- $f(a\cdot b) = f(a) \cdot f(b)$ for all $a,b \in R$
A ring isomorphism is a ring homomorphism which is a bijection. A ring monomorphism (respectively, ring epimorphism) is a ring homomorphism which is an injection (respectively, surjection).
When working in a context in which all rings have a multiplicative identity, one also requires that $f(1_R) = 1_S$ Ring homomorphisms which satisfy this property are called unital ring homomorphisms.
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"ring homomorphism" is owned by djao.
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See Also: ring
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unital, ring isomorphism, ring epimorphism, ring monomorphism, homomorphism, isomorphism, epimorphism, monomprhism |
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Cross-references: property, multiplicative identity, surjection, injection, bijection, function, rings
There are 86 references to this entry.
This is version 7 of ring homomorphism, born on 2001-10-19, modified 2006-10-22.
Object id is 357, canonical name is RingHomomorphism.
Accessed 15982 times total.
Classification:
| AMS MSC: | 13B10 (Commutative rings and algebras :: Ring extensions and related topics :: Morphisms) | | | 16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous) |
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Pending Errata and Addenda
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