We showed in the previous section that every dense subring of the ring of linear transformations of a vector space over a division ring is left primitive. Now, we’d like to prove the converse: every left primitive ring is a dense subring of the ring of linear transformations of some vector space over some division ring. We will assume that is a ring, is a simple left module and As usual, is considered as a right vector space over

**Remark.** If then there exist such that for all we have

*Proof***.** For any let be the -th injection map, i.e. where is the -th coordinate. Clearly Now

**Density Theorem.** (Chevalley – Jacobson) Let be a left primitive ring, a faithful simple left module and Then is a dense subring of

*Proof***.** We have already showed that is a subring of So we need to prove that for any -linearly independent set and any set there exists such that for all The proof is by induction over : if then, since is simple and we have and thus there exists such that Assuming that the result is true for we will have (density condition!). We now prove a claim:

*Claim.* There exists such that and

*Proof of the claim.* Suppose to the contrary that the claim is not true. Then will imply that for any Define by See that is well-defined, that is Therefore, if we put then by the above remark there must exist such that which contradicts -linear independence of

So, using the above claim, for any we can choose such that and for all Thus, since is simple, for all Hence there exist such that Let Then for any

**The Structure Theorem For Primitive Rings**. Let be a left primitive ring, a faithful simple left module and

1) If then

2) If then for any positive integer , there exists a subring of and an *onto* ring homomorphism

*Proof*. It follows from the density theorem and remarks 3 and 4 in the previous post.

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