I was wondering if there is a characterization of rings of weak dimension one. For example, we know that a ring of weak dimension zero is a von Neumann regular ring.
Is there a similar result for rings of weak dimension one?
Thanks for your help :)
2025-01-13 12:01:45.1736769705
Weak dimension of Rings
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Yes, rings with weak dinmension $1$ are the rings with all (three-generated) ideals syzygetic. More precisely, we have the following result:
Theorem: Let $R$ be a commutative ring. Then the following conditions are equivalent:
(1) The weak dimension of $R$ is $\le 1$.
(2) Every ideal of $R$ is of linear type.
(3) Every ideal of $R$ is syzygetic.
(4) Every three-generated ideal of $R$ is syzygetic.
Reference: Rings of weak dimension $1$ and syzygetic ideals, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 124, Number 10, October 1996.