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Bijection Unity Surjective Function Ring Property Injective Function

1	Property
1	Surjective
1	Function
1	Injective Function
1	Bijection
1	Ring
1	Unity

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Ring Homomorphism

Let and be rings. A ring homomorphism is a function $f: R \longrightarrow S$ such that:

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 . Ring homomorphisms which satisfy this property are called unital ring homomorphisms.