# Unit (ring theory)

An element $$x$$ of a non-trivial ringnoteThat 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.

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 $$\mathbb{Z}$$, $$1$$ 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 $$\pm 1$$.

• $$\mathbb{Q}$$ is a field, so every rational except $$0$$ is a unit.

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