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

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

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

\( \phi(0)=0. \)
if \( R \) is a commutative ring, then \( \phi(R) \) is also a commutative ring.
if \( R \) is a field and \( \phi(R) \ne \{0\} \) then \( \phi(R) \) is also a field.
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 \(R\).