Cancellation of projective modules over regular rings with comparability
Let \(R\) be a regular ring, and let \(s\) and \(t\) be positive integers. \(R\) satisfies \((s:t)\)-comparability if for any \(x,y\in R\), either \(t(xR)\lesssim s(yR)\) or \(t(yR)\lesssim s(xR)\); \(R\) satisfies \(s\)-comparability if for any \(x,y \in R\), either \(xR\lesssim s(yR)\) or \(yR\lesssim s(xR)\). It is clear that \((s:t)\)-comparability implies \(s\)-comparability. Open Problem 4 in \textit{K. R. Goodearl}'s book Von Neumann regular rings [Pitman, London (1979; Zbl 0411.16007), Krieger, Malabar, Florida, 2nd ed. (1991; Zbl 0749.16001)]: If \(R\) is a directly finite regular ring which satisfies \((s:t)\)-comparability for some integers \(s \geq t > 0\), is \(R\) unit-regular? This paper answers, in the negative, this problem: a directly finite regular ring with \(s\)-comparability is constructed which is not unit-regular. Moreover, new results on directly finite regular rings with \(s\)-comparability are given. In particular, the authors prove the following result: Let \(R\) be a directly finite regular ring satisfying \(s\)-comparability for some positive integer \(s\). Then for all finitely generated projective right \(R\)-modules \(A\), \(B\), \(C\), \(A\oplus C\cong B\oplus C\) and \(C\lesssim nA\) for some \(n\in\mathbb{N}\Rightarrow A\cong B\).
finitely generated projective right modules
scientific article; zbMATH DE number 896030