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.