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On 2-Von Neumann regular rings. (English) Zbl 1080.13004

Summary: We consider 2-VonNeumann regular rings, that is, rings \(R\) with the property that, if \(F_2\to F_1\to F_0\to E\to 0\) 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 \(m=\infty\), we exhibit a class of 2-VonNeumann 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.

MSC:

13C10 Projective and free modules and ideals in commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
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