# Unit (ring theory)

A unit in a ring is just an element with a multiplicative inverse.

[summary: A unit of a ring is an element with a multiplicative inverse.]

An element $x$ of a non-trivial ring%%note:That is, a ring in which $0 \not = 1$; equivalently, a ring with more than one element.%% 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. %%hidden(Proof): If $xy = xz = 1$, then $zxy = z$ (by multiplying both sides of $xy=1$ by $z$) and so $y = z$ (by using $zx = 1$). %%

# Examples

• In [48l $\mathbb{Z}$], $1$ 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 $\pm 1$.
• [4zq $\mathbb{Q}$] is a field, so every rational except $0$ is a unit.