An element $x$ of a non-trivial ring^[1] is known as a unit if it has a multiplicative inverse: that is, if there is $y$ such that $xy=1$. (We specified that the ring be non-trivial. If the ring is trivial then $0=1$ and so the requirement is the same as $xy=0$; this means $0$ is actually invertible in this ring, since its inverse is $0$: we have $0\times 0=0=1$.)

$0$ is never a unit, because $0\times y=0$ is never equal to $1$ for any $y$ (since we specified that the ring be non-trivial).

If every nonzero element of a ring is a unit, then we say the ring is a field.

Note that if $x$ is a unit, then it has a unique inverse; the proof is an exercise.

Examples

• In $Z$, $1$ and $-1$ are both units, since $1\times 1=1$ and $-1\times -1=1$. However, $2$ is not a unit, since there is no integer $x$ such that $2x=1$. In fact, the only units are $\pm 1$.
• $Q$ is a field, so every rational except $0$ is a unit.

1. ^{^︎}

   That is, a ring in which $0\ne 1$; equivalently, a ring with more than one element.