A short survey of von Neumann regular rings

I’ve talked a lot about von Neumann regular rings on this blog, so I thought I’d write an informal short survey on them, collecting some facts we’ve already seen and many new ones. It should give you an idea of what von Neumann regular rings are. Most of the facts that I did not explicitly cite here are found in the book

Goodearl, K. R. von Neumann regular rings. Second edition. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. xviii+412 pp. ISBN: 0-89464-632-X

The definition

A von Neumann regular ring is an associative ring $A$ such that for each $x\in A$ there exists a $y\in A$ such that $xyx = x$. A popular alternative and equivalent definition is that a ring $A$ is von Neumann regular if and only if every left $A$-module is flat. This is in fact equivalent to every right $A$-module being flat. So we can just drop left and right here.

Examples of von Neumann regular rings

A ring $A$ is von Neumann regular if and only if every $A$-module is flat. So in particular, fields are von Neumann regular. You can also see that using the other definition: if $x$ is a nonzero element in a field, then $xx^{-1}x = x$. If $x = 0$, then any $y$ such as $y=0$ satisfies $xyx = x$.

By the Artin-Wedderburn theorem, finite products of full matrix rings over division rings are von Neumann regular. That’s because So for example, the ring of $2\times 2$ matrices over a field is von Neumann regular.

It’s easy to see that von Neumann regular rings are closed under taking arbitrary products. So infinite products of fields and full matrix rings over division rings are also von Neumann regular. Such examples give von Neumann rings that are not semisimple: for example, in the infinite product $A = \prod_{i=1}^\infty F$ where $F$ is a field, the infinite direct sum $I = \oplus_{i=1}^\infty F$ is such that $A/I$ is flat but not projective. For a proof, see my post on a finitely-generated flat module that is not projective.

Here are some ring constructions and how they can produce von Neumann regular rings:

Theorem. von Neumann regular rings are closed under the following operations:

1. Direct products
2. Direct limits
3. Homomorphic images
4. Centers

All four conditions are suggested exercises for the reader. Another way to get von Neumann regular rings is to take endomorphisms:

Theorem. For any finitely-generated projective module $M$ over a von Neumann regular ring $A$, the endomorphism ring ${\rm End}_A(M)$ is von Neumann regular.

Equivalent conditions

We have already mentioned this somewhat, but here is a fleshing out of the most basic equivalent conditions for a ring to be von Neumann regular.

Theorem. For an associative ring $A$, the following are equivalent:

1. $A$ is von Neumann regular
2. Every $A$-module is flat
3. For every left ideal $I$ of $A$, the left $A$-module $A/I$ is flat
4. For every right ideal $I$ of $A$, the right $A$-module $A/I$ is flat
5. Every principal left (right) ideal is generated by an idempotent
6. Every finitely generated left (right) ideal is generated by an idempotent

Here is the sketch of the proof that a ring is von Neumann regular if and only if every principal ideal is generated by an idempotent. First, if $A$ is von Neumann regular and $xR$ is a principal right ideal, select a $y$ such that $xyx = x$. Then $yx$ is an idempotent and $xyA = xA$. Conversely, suppose every right ideal is generated by an idempotent. Then for any $x\in A$, we have $xA = eA$ for some idempotent $e\in A$. By definition, there exists a $y\in A$ such that $e = xy$, and $y$ is an element such that $xyx = x$.

In the paper

Ming, Roger Yur Chi. On (von Neumann) regular rings. Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 89—91

Ming gave some conditions for a ring to be von Neumann regular. He proved that a reduced ring $A$ is von Neumann regular if and only if every principal left ideal is the left annihilator of some element of $A$. That’s not really for all von Neumann regular rings, though. But he also gave the following equivalent conditions:

Theorem. For an associative ring $A$, the following are equivalent:

1. $A$ is von Neumann regular
2. Every left $A$-module is $p$-injective
3. Every left cyclic $A$-module is $p$-injective

Recall that a left $A$-module $M$ is $p$-injective if for every left principal left ideal $I$ of $A$ and every left $A$-module homomorphism $f:I\to M$, there exists a $y\in M$ such that $f(b) = by$ for all $b\in I$. This is just the usual Baer criterion for injectivity, but restricted to principal ideals. Of course, if every $A$-module were injective, then $A$ would be semismimple, and hence every $A$-module would be projective.

Commutative von Neumann regular rings

Commutative von Neumann regular rings are not as hard to characterize.

Theorem. The following are equivalent for a commutative ring $A$:

1. $A$ is von Neumann regular
2. The localization $A_M$ is a field for all maximal ideals $M$ of $A$
3. $A$ is reduced and has Krull dimension zero
4. All simple $A$-modules are injective

I already talked a little bit about the equivalency of (1) and (3).

Structure theorems for modules

A classic structure theorem for von Neumann regular rings is:

Theorem. If $M$ is a countably generated projective right module over a von Neumann regular ring $A$, then $M$ is a direct sum of modules, each of which is isomorphic to a principal right ideal of $A$.

Unit regular rings and K-Theory

For a ring $A$, let $A^\times$ denote the group of units of $A$, and let $V(A)$ be the subgroup of $A^\times$ generated by the set
$$\{ (ab+1)(ab + 1)^{-1} : ab + 1\in A^\times\}.$$ A von Neumann regular ring $A$ is called unit regular if for each $x\in A$ there exists a unit $y\in A$ such that $xyx = x$. Unit regular rings satsify the cancellation law for finitely-generated projective modules: if $M,N_1,N_2$ are such modules over a unit regular ring, then $M\oplus N_1\cong M\oplus N_2$ implies $N_1\cong N_2$. Another equivalent condition for unit regular is: a von Neumann regular ring is unit regular if and only if whenever $xA + yA = A$, there exists a $z\in A$ such that $x + yz$ is a unit (“stable range 1” condition).

Mencal and Moncasi in

Menal, Pere; Moncasi, Jaume. $K_1$ of von Neumann regular rings. J. Pure Appl. Algebra 33 (1984), no. 3, 295—312.

proved the following:

Theorem. If $A$ is a unit regular von Neumann regular ring then $K_1(A) \cong A^\times/V(A)$.

Going back to unit regular rings, there is a slightly stronger condition for cancellation. A von Neumann regular ring is called separative if for every finitely-generated projective $A$-modules $M$ and $N$, the two conditions

1. $M\oplus M\cong M\oplus N$
2. $M\oplus N\cong N\oplus N$

imply that $M\cong N$ as $A$-modules. In the paper

Ara, P.; Goodearl, K. R.; O’Meara, K. C.; Pardo, E. Diagonalization of matrices over regular rings. Linear Algebra Appl. 265 (1997), 147—163

the authors proved that if $X$ is a square $k\times k$ matrix over a separative von Neumann regular ring, there exists $k\times k$ matrices $Y$ and $Z$ such that $ZXY$ is diagonal.

Let us end with a characterization of unit regular rings given in the paper

Camillo, Victor P.; Khurana, Dinesh. A characterization of unit regular rings. Comm. Algebra 29 (2001), no. 5, 2293—2295.

Here, the authors prove:

Theorem. An associative ring $A$ is unit regular if and only if for every $x\in A$, there exists an idempotent $e\in A$ and a unit $u\in A$ such that $x = e + u$ and $eA\cap xA = \{0\}$.