ring homomorphism
Let R and S be rings. A ring homomorphism is a function f:R⟶S such that:
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f(a+b)=f(a)+f(b) for all a,b∈R
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f(a⋅b)=f(a)⋅f(b) for all a,b∈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(1R)=1S. Ring homomorphisms which satisfy this property are called unital ring homomorphisms.
Title | ring homomorphism |
Canonical name | RingHomomorphism |
Date of creation | 2013-03-22 11:48:50 |
Last modified on | 2013-03-22 11:48:50 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 12 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 13B10 |
Classification | msc 16B99 |
Classification | msc 81P05 |
Related topic | Ring |
Defines | unital |
Defines | ring isomorphism |
Defines | ring epimorphism |
Defines | ring monomorphism |
Defines | homomorphism![]() |
Defines | isomorphism![]() |
Defines | epimorphism![]() |
Defines | monomprhism |