Weak dimension one rings are axiomatizable

Let $R$ be a ring. In the previous post on pure exact sequences, we called an exact sequence $0\to A\to B\to C\to 0$ of left $R$-modules pure if its image under any functor $X\otimes -$ is an exact sequence of abelian groups for any right $R$-module $X$. Here is yet another characterization of purity:

Theorem. Let $R$ be a ring. An exact sequence of $R$-modules $0\to A\to B\to C\to 0$ is pure if and only if for every diagram
there exists an $R$-module homomorphism $\theta:R^m\to A$ such that $\theta\sigma=\alpha$.

I won't prove this here since an excellent exposition can be found in

Lam, T. Y. Lectures on modules and rings. Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999. xxiv+557 pp. ISBN: 0-387-98428-3

I've recommended this book before as it is one of the top three books on modules and rings in existence. The proof can be found on page 155. So, why are we even mentioning this characterization of pure exact sequences? Well, it has something to do with axiomatization! Let's see how.

Weak Dimension

Recall that the flat dimension of a left $R$-module $M$ is the infimum over lengths of flat resolutions of $M$, and the weak dimension of $R$ is the supremum of flat dimensions of all left $R$-modules. Notice that we used "left" in this definition, but the weak dimension is the same when right $R$-modules are used, which is different than the case of global dimension.

The rings of weak dimension zero are the von Neumann regular rings. These are precisely the rings $R$ such that for every $x\in R$ there exists a $y\in R$ such that $xyx = x$. Under this characterization, the rings of weak dimension zero can be axiomatized in first order logic: they are exactly those rings that satisfy the sentence $\forall x\exists y(xyx = x)$.

The whole point of my foray into the realm of pure exact sequences was so that we could answer the following question: are the rings of weak dimension one axiomatizable in first-order logic? Yes!

Weak Dimension One

To prove this, it suffices to show that the rings of weak dimension at most one can be so axiomatized, since the rings of weak dimension zero can be axiomatized. Now, it is easy to see that the following result is true.

Proposition. A ring has weak dimension at most one if and only if for every natural number $n$, every ideal generated by at most $n$ elements is flat.

Therefore, in order to show that the rings of weak dimension at most one are axiomatizable, it suffices to show that for each natural number, there exists a first-order sentence saying that every $n$-generated ideal is flat. We do this via purity: we show that there is a first-order sentence that says that for every $n$-generated ideal $I$, every exact sequence $0\to A\to B\to I\to 0$ is pure. Recall the theorem we had at the beginning:

Theorem. Let $R$ be a ring. An exact sequence of $R$-modules $0\to A\to B\to C\to 0$ is pure if and only if for every diagram

there exists an $R$-module homomorphism $\theta:R^m\to A$ such that $\theta\sigma=\alpha$.

Using this theorem, we get the following:

Corollary. Let $R$ be a ring. Then every left $n$-generated ideal is flat if and only if for every homomorphism $f:R^n\to R$ of left $R$-modules fitting into the diagram
such that $f\alpha = 0$, there exists a homomorphism $\theta:R^m\to R^n$ of left $R$-modules such that $\theta\sigma = \alpha$ and such that $f\theta = 0$.

The existence of $\theta$ is not vacuous. Although $\beta$ is nearly like $\theta$ in that $\beta\sigma = \alpha$ and hence $f\beta\sigma = 0$, it is not guaranteed that $f\beta = 0$. Now notice that in this corollary, we have omitted mentioning the kernel of $f$ as an explicit object. That is because we only want to ever speak of homomorphisms of free modules of fixed finite rank, which can be put into a first-order sentence. Hence, we have:

Corollary. The class of all rings of weak dimension one is axiomatizable in first-order logic.