# Ideals of Field

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## 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

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 23$: Theorem $23.5$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2$: Exercise $12$