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

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

By definition, a field is a division ring.

From Ideals of Division Ring, it follows that the only ideals of $\struct {R, +, \circ}$ are $\struct {R, +, \circ}$ and $\set {0_R}$.

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

Let $a \in R^*$.

Then $\ideal a$ is a non-null ideal and hence is $R$.

Thus $1_R \in \ideal a$.

Thus $\exists x \in R: x \circ a = 1_R$ by the definition of principal ideal.

Therefore $a$ is invertible.

Thus by definition $a$ is a division ring such that $\circ$ is commutative.

The result follows by definition of field.