**Fact 1. **Let be a left primitive ring and a faithful simple left module. By Schur’s lemma is a division ring and can be considered as a right vector space over in the usual way. Let and define by for all and Then is a well-defined ring homomorphism. Also is one-to-one because is faithful. So can be viewed as a subring of

**Fact 2**. Every left primitive ring is prime. To see this, suppose is a faithful simple left module and be two non-zero ideals of with Now is a submodule of and is simple. Therefore either or If then we get which is nonsense. Finally, if then we will have Thus and so a contradiction!

**Fact 3**. A trivial result of Fact 2 is that the center of a left primitive ring is a commutative domain. A non-trivial fact is that every commutative domain is the center of some left primitive ring. For a proof of this see: T. Y. Lam, A First Course in Noncommutative Ring Theory, page 195.

**Fact 4**. Let be a prime ring and a faithful left module of finite length. Then is left primitive. To see this, let be a composition series of Therefore is a simple left module for every We also let Then each is an ideal of and it’s easy to see that Thus because is faithful. Hence for some because is prime. Therefore is a faithful simple left module.

**Fact 5**. Every left primitive ring is semiprimitive. This is easy to see: let be a faithful simple left module and , as usual, be the Jacobson radical of The claim is that . So suppose that and choose Then because is simple, and so Also either , which is impossible because then , or If then for some Thus which gives us the contradiction because is invertible in