< back to index

Unit

Definition

Given a ring $R$, an element $r \in R$ is a unit if there exists $s \in R$ such that $rs = 1 = sr$. We say $s$ is the inverse of $r$ and write $s = r^{-1}$.

Two elements $a,b \in R$ are said to be associates if there exists a unit $r \in R^\times$ such that $ar = b$. This defines an equivalence relation.

Criterion

In an integral domain, and (allegedly) in any Noetherian ring, it is sufficient to show that there exists a one-sided inverse e.g. $sr=1$, and it follows that also $rs=1$.

TODO find a counterexample (it will have to be an infinite, non-commutative, non-noetherian ring!)

Also, existence of both a left- and right- inverse implies they are equal.

Properties

A unit is never a zero-divisor. (Proof: suppose $a$ is a unit, then $ab=0 \implies a^{-1}ab=0 \implies b=0$.)

The set of units is a group, denoted $R^\times$.

all these pages adapted with probably insufficient credit from my university's lecture notes