Unit of a ring

If \(u\) is a unit, its inverse \(v\) is unique and is written \(u^{-1}\). The elements \(1\) and \(-1\) are always units, and zero never is except in the trivial ring. The product of two units is a unit, since \(uv\) has inverse \(v^{-1}u^{-1}\), and the inverse of a unit is also a unit; this is why the set \(R^\times\) of units is a group under multiplication. The sum of units, however, need not be a unit.

Guiding examples: in \(\mathbb{Z}\) the units are \(\pm1\); in \(\mathbb{Z}_n\), the \(a\) with \(\gcd(a,n)=1\), and their number is \(\varphi(n)\) (for instance, \(\mathbb{Z}_{12}\) has four); in \(\mathbb{Z}[i]\) they are \(\pm1,\pm i\); and in \(\mathbb{Z}[\sqrt2]\) there are infinitely many, such as the powers of \(1+\sqrt2\). A field is exactly a commutative ring in which every nonzero element is a unit.

A typical mistake is to confuse «unit» with the identity element \(1\). The identity is one particular unit, but a ring can have many more. Another mistake is to think that an element with no inverse is always a zero divisor; in \(\mathbb{Z}\), \(2\) is not a unit and it is not a zero divisor either. That difference is studied in the next concept, the zero divisor.

Units underlie later notions: two elements are associates when they differ by a unit, an irreducible element cannot be factored except with units, and unique factorization is stated «up to units». So it is worth knowing how to compute them in each concrete ring.