Let $R$ be a ring with multiplicative identity $1$ . We say that $u\in R$ is an unit (or unital) if $u$ divides $1$ (denoted $u \mid 1$ ). That is, there exists an $r\in R$ such that $1=ur=ru$ .

Notice that $r$ will be the multiplicative inverse (in the ring) of $u$ , so we can characterize the units as those elements of the ring having multiplicative inverses.

In the special case that $R$ is the ring of integers of an algebraic number field $K$ , the units of $R$ are sometimes called the algebraic units of $K$ (and also the units of $K$ ). They are determined by Dirichlet's unit theorem.