# ring homomorphism

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.

 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