ring homomorphism

Let $R$ and $S$ be rings. A ring homomorphism is a function $f:R⟶S$ 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