**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.