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

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.5$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2$: Exercise $12$