Throughout is a division ring, a right vector space over and is a subring of Clearly can be viewed as a left module by defining for all and .

**Definition.** is said to be a **dense** **subring** of if for every -linearly independent set and any set there exists such that for all

**Remark 1.** If is a dense subring of then is left primitive.

*Proof*. Well, is clearly a faithful left module. To see why it is simple, let and By the density condition, there exists such that Thus

**Remark 2**. We proved in example 4 in here that itself is left primitive. In the above remark we showed that any dense subring of is also left primitive.

**Remark 3**. If is a dense subring of and then

*Proof*. Let be a basis for as a vector space over and let Then, since is dense, there exists such that for all . But then and hence

**Remark 4**. If is a dense subring of and then for any positive integer , there exists a subring of and an onto ring homomorphism

*Proof*. Let be a -linearly independent subset of and let i.e. the -vector subspace of spanned by Now define Clearly is a subring of . Finally we define the map by for all and It’s easy to see that is a well-defined ring homomorphism. To prove that is onto, let Then, by the density condition, there exists such that for all Thus because and clearly

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