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.

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.