Unit in a ring

A unit in a ring is an element with a multiplicative inverse.

If $z$ is a nilpotent element of a ring, choose $m\ge 1$ with $z^m=0$. The finite geometric series identity givesand the factors commute, so $1-z$ is a unit. In a commutative ring, every multiple of a nilpotent element is again nilpotent. Consequently an equality $x=a{x}^2$ with $x$ nilpotent forces $x=0$: rewrite it as $(1-ax)x=0$ and multiply by the inverse of $1-ax$.

A nonunit is an element of a ring that has no multiplicative inverse in that ring.

The units of a ring $R$ form a group $R^{\times }$ under multiplication.

A local unit has valuation zero in the local field, or equivalently is an invertible element of its valuation ring. A principal unit additionally reduces to one in the residue field. The distinction between local units and principal units retains the prime-to-p residue roots of unity in a cyclotomic p-adic field.

The unit signature map records the signs of a unit under all real field embeddings. Its kernel consists of totally positive units. Its image determines the kernel of the homomorphism from the narrow ideal class group to the ordinary ideal class group: this kernel is $\{\pm 1{\}}^{r_1}/\mathrm{sgn}({\mathcal{O}}_K^{\times })$.

For a number field $K$ and a finite set $S$ of finite primes, an S-unit is an element of $K^{\times }$ whose discrete valuation is zero at every prime outside $S$. Its group is the unit group of the ring obtained by allowing denominators at $S$. The Dirichlet unit theorem and the valuation homomorphism to ${\mathbb{Z}}^S$ show that this group is a finitely generated abelian group.

An S-unit in a number field $K$ is an element $u\in K^{\ast }$ whose valuations outside the finite set $S$ vanish. Equivalently, it is a unit of the ring of S-integers ${\mathcal{O}}_{K,S}$. The S-unit group is finitely generated by the Dirichlet unit theorem together with generators arising from the inverted primes.

The multiplicative group of integers modulo $n$ consists of the residue classes coprime to $n$. It has order $\phi (n)$.