Field

Thus, under the convention used here that a ring need not have a multiplicative identity, the unital condition is part of the definition of a field.

For a commutative unital ring $R$ with $1\neq 0$, the following are equivalent: $R$ is a field; its only ideals are $(0)$ and $R$; and $(0)$ is maximal. The hypotheses matter: a nonunital ring can have only two ideals without being a field.

• $\mathbb{Q}$ and $\mathbb{R}$ are fields.
• For a prime $p$, $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ is a field.
• $\mathbb{Z}$ is not a field since $2$ has no inverse in $\mathbb{Z}$.