ring homomorphism

Let R and S be rings. A ring homomorphismMathworldPlanetmath is a function f:RS such that:

  • f(a+b)=f(a)+f(b) for all a,bR

  • f(ab)=f(a)f(b) for all a,bR

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 identityPlanetmathPlanetmath, 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 homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath
Defines isomorphismMathworldPlanetmathPlanetmath
Defines epimorphismMathworldPlanetmathPlanetmath
Defines monomprhism