Question

Let A be a commutative ring with unity and let F be a field. Let f be a homomorphism from A onto F. Prove that ker F is a maximal ideal in A.

Answer 1

Conditions

Let A be a commutative ring with unity and let F be a field. Let f be a homomorphism from A onto F. Prove that $\ker f$ is a maximal ideal in A.

Solution

As A is a commutative ring with unity, then:

$$\exists e \in A: \forall a \in A \ a \cdot e = e \cdot a = a$$

And

$$\forall a, b \in A \ a \cdot b = b \cdot a$$

As F is a field, then it's a commutative ring with unity which is not equal to 0.

$$f: A \to F: \forall a, b \in A \ f(a \cdot b) = f(a) \cdot f(b)$$ $$\ker(f) = \{x \in A: f(x) = 0\}$$

We know a theorem: M is a maximal ideal $\Leftarrow \Rightarrow$ A/M is a field.

And as the F is field, then $A / \ker(f)$ is a field.

That's why $\ker(f)$ is a maximal ideal.