# 6.2: Ring Homomorphisms

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### Ring Homomorphisms

##### Definition: Ring homomorphisms

Let $$R, S$$ be rings. A function $$\phi: R \to S$$ is called a ring homomorphism if

1. $$\phi(a+b)=\phi(a)+\phi (b), \forall a,b \in R,$$ and

2. $$\phi(ab)=\phi(a) \phi (b), \forall a,b \in R.$$

If $$\phi$$ is bijective, then $$\phi$$ is called isomorphism.

##### Definition: Kernel of a homomorphism

The kernel of $$\phi$$ is denoted and defined as $$Ker(\phi) = \{ r \in R | \phi(r)=0 \}.$$

##### Example $$\PageIndex{1}$$

Define $$\phi: \mathbb{Z} \to \mathbb{Z}_n$$ by $$\phi(a)=a(mod n), \forall a\in R.$$

Then $$\phi$$ is a ring homomorphism but not an isomorphism. In this case, $$Ker(\phi) = n \mathbb{Z}.$$

## Properties of ring homomorphisms:

##### Theorem $$\PageIndex{1}$$

Let $$R, S$$ be rings. Let $$\phi: R \to S$$ be a ring homomorphism. Then,

Then

1. $$\phi(0)=0.$$

2. if $$R$$ is a commutative ring, then $$\phi(R)$$ is also a commutative ring.

3. if $$R$$ is a field and $$\phi(R) \ne \{0\}$$ then $$\phi(R)$$ is also a field.

4. if $$R$$ and $$S$$ have an identity. If $$\phi$$ is onto then $$\phi(1_R)=1_S.$$

## Ideals:

##### Definition: Ideal

Let $$R$$ be a  ring. Then  a subring $$I$$ of $$R$$ is called an ideal of $$R$$ if $$ar \in I$$, and $$ra \in I, \forall a \in I$$ and $$r \in R.$$

##### Theorem $$\PageIndex{2}$$

Let $$R, S$$ be rings. Let $$\phi: R \to S$$ be a ring homomorphism. Then $$Ker (\phi)$$ is an ideal of $$R$$.

This page titled 6.2: Ring Homomorphisms is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.