Let
R be a
ring. An element
u of
R is called a
unit
if it has a multiplicative
inverse. That is,
there exists
v in
R such that
uv=1 and
vu=1.
If u is a unit then the v in the definition is unique.
For if uw=1 then vuw=v and since vu=1 this says
that w=v. The inverse of a unit u is usually denoted
by u-1.
The collection of all units of R form a group, the
multiplicative group of R.
Examples
-
1,-1 are the only units in the ring of integers Z.
-
In a field every nonzero element is a unit.
-
An nxn matrix with complex number coefficients is a unit in
the ring of all such matrices if and only if it has nonzero determinant.