Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points.
An integral domain is a commutative ring with the property that if and
, then
. Hence, if
, then
or
(or both).
The characteristic of an integral domain is the lowest positive integer such that
.
Let . Then
. This is because
.
If , then we have
. This is obvious for
. If
, then this implies
, which contradicts the fact that
is the lowest positive integer such that
added
times to itself is equal to
. Hence, if
is the characteristic of the integral domain
, then it is the lowest positive integer such that any non-zero member of
, added
times to itself, gives
. No member of
can be added a lower number of times to itself to give
.
Sometimes is written as
. One should remember that this has nothing to do the multiplication operator in the ring. In other words, this does not imply that
, where
is a member of the domain. In fact,
does NOT have to be a member of the domain. It is just an arbitrary positive integer.
Now on to an important point: something that is not emphasized, but should be. Any expression of the form
.
Now use this knowledge to prove that the characteristic of an integral domain, if finite, has to be or prime.