# Ideals of Field

## Theorem

Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Then $\struct {R, +, \circ}$ is a field if and only if the only ideals of $\struct {R, +, \circ}$ are $\struct {R, +, \circ}$ and $\set {0_R}$.

## Proof

### Necessary Condition

Let $\struct {R, +, \circ}$ be a field.

The result follows from Field has 2 Ideals.

$\Box$

### Sufficient Condition

Suppose that the only ideals of $\struct {R, +, \circ}$ are $\struct {R, +, \circ}$ and $\set {0_R}$.

The result follows from Commutative and Unitary Ring with 2 Ideals is Field

$\blacksquare$