IBN and related properties for rings
Let \(R\) be an associative ring with \(1\neq 0\). Then \(R\) has invariant basis number (IBN) if \(R^m\cong R^n\) in \(R\)-mod implies that \(m=n\), and \(R\) has the rank condition if \(R^m\to R^n\to 0\) exact in \(R\)-mod implies that \(m\geq n\). If \(R=R_1\times R_2\) then (1) \(R\) fails to have IBN if and only if each \(R_i\) fails to have IBN, and (2) \(R\) fails to have the rank condition if and only if each \(R_i\) fails to have the rank condition. A formal \(2\times 2\) block triangular matrix ring \(T\) has IBN (rank condition) if and only if one of the diagonal blocks has IBN (rank condition); \(T\) is stably finite if and only if both of its diagonal blocks are stably finite. Stable finiteness is a Morita invariant property.
scientific article; zbMATH DE number 1912016