ring homomorphism
Let $R$ and $S$ be rings. A ring homomorphism^{} is a function $f:R\u27f6S$ such that:

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$f(a+b)=f(a)+f(b)$ for all $a,b\in R$

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$f(a\cdot b)=f(a)\cdot f(b)$ 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.
Title  ring homomorphism 
Canonical name  RingHomomorphism 
Date of creation  20130322 11:48:50 
Last modified on  20130322 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 