Noetherian rings of injective dimension one which are orders in quasi-Frobenius rings. (Q1419011)

The paper deals with Noetherian orders \(R\) in a quasi-Frobenius ring \(Q\). It is shown first that \(\text{id}\, R_R=1\) implies that \(R\) has Krull dimension \(\leq 1\) and that \(R/I\) is quasi-Frobenius for every invertible ideal \(I\) of \(R\). Conversely, it is proved that if \(\text{Kdim}\, R=1\) and every maximal right ideal of \(R\) contains an invertible ideal \(I\) with \(R/I\) quasi-Frobenius, then \(\text{id}\, R_R=1\). If \(R\) is an indecomposable Noetherian ring with \(\text{id}\, R_R=1\) which is integral over its center, the authors show that \(R\) is an order in a quasi-Frobenius ring if and only if the Artinian radical of \(R\) is zero. Some criteria for injective dimension one are also given in case \(R\) is semiprime.