Notation. Throughout this post we will assume that is a field,
is a
-vector space,
and
Obviously
is a two-sided ideal of
If then
the ring of
matrices with entries in
and thus
is a simple ring, i.e. the only two-sided ideals of
are the trivial ones:
and
But what if
What can we say about the two-sided ideals of
if
Theorem 1. If is countably infinite, then
is the only non-trivial two-sided ideal of
Proof. Let be a two-sided ideal of
and consider two cases.
Case 1. So there exists
such that
Let
be a basis for
and let
be a subspace of
such that
Note that
is also countably infinite dimensional because
Let
be a basis for
Since
the elements
are
-linearly independent and so we can choose
such that
for all
Now let
be such that
for all
Then
and so
Case 2. Choose
and suppose that
Let
be a basis for
and extend it to a basis
for
Since
there exists
such that
Let
and fix an
such that
Now let
and suppose that
Let
be a basis for
and for every
put
For every
define
as follows:
and
for all
and
for all
See that
and so
Exercise. It should be easy now to guess what the ideals of are if
is uncountable. Prove your guess!
Definition. Let be an integer. A ring with unity
is called
-simple if for every
there exist
such that
Remark 1. Every -simple ring is simple. To see this, let
be a two-sided ideal of
and let
Then, by definition, there exist
such that
But, since
is a two-sided ideal of
we have
for all
and so
It is not true however that every simple ring is -simple for some
For example, it can be shown that the first Weyl algebra
is not
-simple for any
Theorem 2. If then
is
-simple. If
is countably infinite, then
is
-simple.
Proof. If then
and so we only need to show that
is
-simple. So let
and suppose that
is the standard basis for
Since
there exists
such that
Using
it is easy to see that
where
on the right-hand side is the identity matrix. This proves that
is
-simple. If
is countably infinite, then, as we proved in Theorem 1, for every
there exist
such that
That means
is
-simple.
Remark 2. An -simple ring is not necessarily artinian. For example, if
is countably infinite, then the ring
is
-simple but not artinian.