We saw in part (2) that von Neumann regular rings live somewhere between semisimple and semiprimitive rings. The goal in this post is to prove a theorem of Armendariz and others which gives a necessary and sufficient condition for a ring to be both regular and reduced. This result extends Kaplansky’s result for commutative rings (see the corollary at the end of this post). We remark that a commutative von Neumann regular ring is necessarily reduced. That is because if for some then choosing with we will get

**Definition **. A von Neumann regular ring is called **strongly regular** if is reduced.

**Theorem 1.** (Armendariz, 1974) A ring with 1 is strongly regular if and only if is a division ring for all maximal ideals of

*Proof*. Suppose first that is strongly regular and let be a maximal ideal of Let So for all Since is regular, there exists some such that Then is an idempotent and thus because in a reduced ring every idempotent is central. Since we have and hence Thus is a right inverse of Similarly and is a left inverse of Therefore is invertible and hence is a division ring. Conversely, suppose that is a division ring for all maximal ideals of If is not reduced, then there exists such that

Let Clearly is a proper ideal of and hence for some maximal ideal of But then in which is a division ring. Thus i.e. there exists some such that which is absurd. To prove that is von Neumann regular, we will assume, to the contrary, that is not regular. So there exists such that for all Let Clearly is a proper ideal of and so for some maximal ideal of It is also clear that if for some then because we may choose Thus in and hence there exists some and such that Therefore for some But then and so which is nonsense. This contradiction proves that must be regular.

**Corollary**. (Kaplansky) A commutative ring is regular if and only if is a field for all maximal ideals of

At the end let me mention a nice property of strongly regular rings.

**Theorem 2**. (Pere Ara, 1996) If is strongly regular and for some then is a unit for some

how about elementary divisor rings?

To add some more context to Theorem 2, that a ring with that property (that if , then there exists such that is a unit) is said to have “stable range 1.” It is indeed an interesting proprety.

The citation for the theorem doesn’t fit well though. In that paper, Pere proves that strongly *PI* regular rings have stable range 1, which is much more amazing 🙂

Strongly regular rings were known to have this property earlier than 1996. (For example, it is already in Goodearl’s 1991 text on von Neumann regular rings, and for all I know it appeared much earlier than that even.)

Well, strongly regular rings are strongly -regular and so the theorem in my post is just a special case of Pere’s theorem.