# unit

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.

 Title unit Canonical name Unit Date of creation 2013-03-22 11:56:28 Last modified on 2013-03-22 11:56:28 Owner drini (3) Last modified by drini (3) Numerical id 15 Author drini (3) Entry type Definition Classification msc 16B99 Synonym unital Related topic Associates Related topic Prime Related topic Ring Related topic UnitsOfQuadraticFields Defines algebraic unit