Fields and trivial ideals

Let $R$ be a nonzero commutative ring with identity. Then $R$ is a field if and only if its only ideals are $(0)$ and $R$ . Equivalently, every nonzero element of $R$ is a unit.

Proof idea

In a field, an ideal containing $a\ne 0$ also contains $a^{-1}a=1$ , so it is all of $R$ . Conversely, if the only ideals are $(0)$ and $R$ , then $(a)=R$ for every $a\ne 0$ , so $ab=1$ for some $b\in R$ .

This criterion is often paired with maximal iff quotient is a field.