On 2-von Neumann Regular Rings

In this article, we consider 2-von Neumann regular rings, that is, rings R with the property that, if F 2 → F 1 → F 0 → E → 0 is an exact sequence of R-modules with F 0, F 1, and F 2 finitely generated free modules, then the module E is projective. For each positive integer m, as well as for m = ∞, we exhibit a class of 2-von Neumann regular rings with Krull dimension m. For this purpose, we study trivial extensions of local rings by infinite-dimensional vector spaces over their residue fields. The article includes a brief discussion of the scope and precision of our results.