An element 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 $x y = 1$ . (We specified that the ring be non-trivial. If the ring is trivial then $0 = 1$ and so the requirement is the same as $x y = 0$ ; 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.

Proof

If $x y = x z = 1$ , then $z x y = z$ (by multiplying both sides of $x y = 1$ by $z$ ) and so $y = z$ (by using $z x = 1$ ).

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 $2 x = 1$ . In fact, the only units are $\pm 1$ .

$Q$ is a field, so every rational except $0$ is a unit.

^{^︎}
That is, a ring in which $0 \neq 1$ ; equivalently, a ring with more than one element.