Definition. A ring is called von Nemann regular, or just regular, if for every there exists such that
Remark 1. Regular rings are semiprimitive. To see this, let be a regular ring. Let the Jacobson radical of and choose such that Then and, since is invertible because is in the Jacobson radical of we get
Examples 1. Every division ring is obviously regular because if then for all and if then for
Example 2. Every direct product of regular rings is clearly a regular ring.
Example 3. If is a vector space over a division ring then is regular.
Proof. Let and There exist vector subspaces of such that So if then for some unique elements and We also have for some unique elements and Now define by It is obvious that is well-defined and easy to see that and
Example 4. Every semisimple ring is regular.
Proof. For a division ring the ring is regular by Example 3. Now apply Example 2 and the Wedderburn-Artin theorem.
Theorem. A ring is regular if and only if every finitely generated left ideal of is generated by an idempotent.
Proof. Suppose first that every finitely generated left ideal of can be generated by an idempotent. Let Then for some idempotent That is and for some But then Conversely, suppose that is regular. We first show that every cyclic left ideal can be generated by an idempotent. This is quite easy to see: let be such that and let Clearly is an idempotent and Thus and so Also and hence So and we’re done for this part. To complete the proof of the theorem we only need to show that if then there exists some idempotent such that To see this, choose an idempotent such that Thus Now choose an idempotent such that and put See that is an idempotent, and Thus Let Then is an idempotent and
Corollary. If the number of idempotents of a regular ring is finite, then is semisimple.
Proof. By the theorem, has only a finite number of left principal ideals. Since every left ideal is a sum of left principal ideals, it follows that has only a finite number of left ideals and hence it is left Artinian. Thus is semisimple because is semiprimitive by Remark 1.
Remark 2. The theorem is also true for finitely generated right ideals. The proof is similar.
Remark 3. Since, by the Wedderburn-Artin theorem, a commutative ring is semisimple if and only if it is a finite direct product of fields, it follows from the Corollary that if the number of idempotents of a commutative von Neumann regular ring is finite, then is a finite direct product of fields.