5.3: Ring Homomorphisms

 

Definition: Ring homomorphisms

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

1. Preservation of addition: \( \phi(a+b)=\phi(a)+\phi (b), \forall a,b \in R, \) and 

2. Preservation of multiplication: \( \phi(ab)=\phi(a) \phi (b), \forall a,b \in R. \)

 

If \( \phi \) is bijective, then \( \phi \) is called isomorphism.

Throughout this course, a ring homomorphism is a function that preserves addition and multiplication. When rings have multiplicative identities, we will explicitly say unital ring homomorphism if preservation of the identity is also required.

Example \(\PageIndex{1}\)

Show that  the function \(\phi:\mathbb{Z}_2 \to \mathbb{Z}_2\) defined by \(\phi(x)=x^2, x \in \mathbb{Z}_2 \) is a ring homomorphism.

Solution

Let \(a, b \in \mathbb{Z}_2.\)

Consider \( \phi(a+b)=(a+b)^2=a^2+2ab+b^2=a^2+b^2=\phi(a)+\phi (b).\)

Consider \( \phi(ab)=(ab)^2=a^2b^2=\phi(a)\phi (b).\)

Hence, \(\phi\) is a ring homomorphism.

Since \( \phi(1)=1^2=1\), \(\phi\) is a unital  ring homomorphism.

 

Example \(\PageIndex{2}\)

Whether  or not the function \(\phi:\mathbb{Z} \to \mathbb{Z}\) defined by \(\phi(x)=2x, x \in \mathbb{Z} \) is a ring homomorphism? Justify your answer.

Solution

Choose \(a=2\) and \(b=3\). Then   \( \phi(2)=  4\) and \(\phi(3)=6\),  \(\phi(a)\phi (b)=(4)(6)=24\) but \( \phi(ab)=\phi((2)(3))=\phi(6)=12\ne \phi(a)\phi(b)\). 

Hence, \(\phi\) is not a ring homomorphism.

Definition: Kernel of a homomorphism

Let \( R, S \) be rings. Let \( \phi: R \to S \) be a ring homomorphism. Then the kernel of \( \phi \) is denoted and defined as \( \ker(\phi) = \{ r \in R \mid \phi(r)=0_S \}. \)

Definition: Image

Let \( R, S \) be rings. Let \( \phi: R \to S \) be a ring homomorphism. Then the Image of \( \phi \) is denoted and defined as \( Im(\phi) = \{  \phi(r) \in S \mid r \in R\}\subset S. \)

Example \(\PageIndex{3}\)

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

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. \( \phi(0_R)=0_S. \)

2. \( \phi(-a)=- \phi(a), \forall a\in R.\)

3. \( \phi(a^n)=\left(\phi(a)\right)^n, \forall a\in R, n \in \mathbb{Z}_+\)

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

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

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

7. Let \( R, S \) be rings. Let \( \phi: R \to S \) be a ring homomorphism. Then \( \ker (\phi)\) is an ideal of \(R\) and \(Im (\phi)\) is an ideal of \(S\).

Theorem \(\PageIndex{2}\)

Let \( R, S \) be rings and let \( \phi: R \to S \) be a ring homomorphism. 

\(\phi\) is injective if and only if \(\ker(\phi)=\{0_S\}\).

Theorem \(\PageIndex{3}\)

Let \(R\) be ring  and let \(I\) be an ideal \(R\), then the map \(\pi:R\to R/I\) defined by $\(pi(a)=a+I\) is a surjective ring homomorphism and \(\ker(\pi)=I\).

Example \(\PageIndex{4}\)

A quotient of \(\mathbb{Z}_{12}\):
Let \(I=\{[0],[4],[8]\}\). Then \(I\lhd \mathbb{Z}_{12}\). Its cosets are
\begin{align*}
I&=\{[0],[4],[8]\}, & [1]+I&=\{[1],[5],[9]\},\\
[2]+I&=\{[2],[6],[10]\}, & [3]+I&=\{[3],[7],[11]\}.
\end{align*}
The map \(\phi:\mathbb{Z}_{12}\to \mathbb{Z}_4, \phi([a]_{12})=[a]_4\), is a well-defined surjective ring homomorphism with kernel \(I\). Therefore, by the first isomorphism theorem,
\(
\boxed{\mathbb{Z}_{12}/I\cong\mathbb{Z}_4.}
\)