[summary: A unit of a ring is an element with a multiplicative inverse.]
An element of a non-trivial ring%%note:That is, a ring in which ; equivalently, a ring with more than one element.%% is known as a unit if it has a multiplicative inverse: that is, if there is such that . (We specified that the ring be non-trivial. If the ring is trivial then and so the requirement is the same as ; this means is actually invertible in this ring, since its inverse is : we have .)
is never a unit, because is never equal to for any (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 is a unit, then it has a unique inverse; the proof is an exercise. %%hidden(Proof): If , then (by multiplying both sides of by ) and so (by using ). %%
Examples
- In [48l ], and are both units, since and . However, is not a unit, since there is no integer such that . In fact, the only units are .
- [4zq ] is a field, so every rational except is a unit.