Rings in this question are assumed to be commutative. I am asking "natural" examples of rings of weak dimension $\le1$ which are not semihereditary. ~~It would be better if there are integral domains as examples.~~ (see Update)

There are several equivalent characterizations of rings of weak dimension $\le1$ in [SP, Tag 092S]. One is the following: a ring is of weak dimension $\le1$ if every finitely generated ideal is flat.

Another slightly stronger concept is that of semihereditary rings. When talking about integral domains, they are also called Prüfer. A ring is semihereditary if every finitely generated ideal is projective.

Since a module is finite projective if and only if it is flat of finite presentation, we see that a ring of weak dimension $\le1$ is semihereditary if and only if it is coherent. However, I want to see some examples of non-coherent rings of weak dimension $\le1$, especially those which appear "naturally" in commutative algebra or algebraic geometry. A bit googling does not lead me to anything.

**Update:** There is no such example for integral domains: every finitely generated flat module over an integral domain is necessarily projective. More generally, there is a criterion for semihereditary rings in

Vasconcelos, W. V. (1969). *On Finitely Generated Flat Modules.* Transactions of the American Mathematical Society, 138, 505–512. https://doi.org/10.2307/1994928

Theorem 4.2.For a commutative ring $R$, the following are equivalent:

- $R$ is semihereditary;
- $R$ has weak dimension at most one and the annihilator of each element is finitely generated.

This theorem suggests that there are examples of rings of weak dimension $\le1$ but not semihereditary. However, I do not find any example in the literature. On the other hand, it seems slightly counterintuitive that, rings of weak dimension $0$ (aka. absolutely flat rings, or von Neumann regular rings) are semihereditary, since every finitely generated ideal is a direct summand of the ring itself, thus projective.

blunderson this site resulting from users blindly inserting words into template statements without realising which sections have "all rings are Noetherian" implicitly written somewhere. $\endgroup$