16B99-Unit.tex

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\begin{document}
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 {\em algebraic units} of $K$ (and also the units of $K$).\, They are determined by Dirichlet's unit theorem.
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