# 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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