On self-injective dimensions of Artinian rings
Let \(R\) be a left and right artinian ring and \(\text{mod }R\) the category of finitely generated left \(R\)-modules. The author studies the question when \(\text{inj dim }R_ R < \infty\) implies \(\text{inj dim }_ RR < \infty\). His main result is the following: Let \(m, n \geq 1\). Suppose that \(\text{inj dim }R_ R \leq n\) and that for all \(0 \leq i \leq n - 2\) \(\text{Ext}^ i_ R (\text{Ext}^ m_ R (-,R), R)\) vanishes on \(\text{mod }R\). Then \(\text{inj dim }_ RR < \infty\).
category of finitely generated left modules
scientific article; zbMATH DE number 701543