Unit

Let $R$ be a unital ring . An element $u\in R$ is a unit if there exists $v\in R$ such that $uv=vu=1$.

Units control invertibility of scalars in algebraic constructions and determine the group of units . In commutative rings, units are precisely the elements that generate the whole ring as a principal ideal.

Examples:

• The units of $\mathbb Z$ are $\pm 1$.
• In a field $k$, every nonzero element is a unit.
• In $\mathbb Z$, the element $2$ is not a unit.