Tiled orders and Frobenius rings. (Q1810209)

Let \(R\) be a discrete valuation ring with quotient field \(K\) and \({\mathfrak p}:=\text{Rad\,}R\). For an \(R\)-order \(\Lambda\) in a symmetric \(K\)-algebra \(A\), the \((\Lambda,\Lambda)\)-bimodules \(\Lambda\) and \(\Lambda^*:=\hom_R(\Lambda,R)\) can be regarded as full \(R\)-lattices in \(A\). Choose any \(n\in\mathbb{N}\) such that \({\mathfrak p}^n\Lambda^*\subset\text{Rad}^2\Lambda\) in \(A\). Then \(\Lambda/{\mathfrak p}^n\Lambda^*\) is a symmetric Artinian algebra with the same Gabriel quiver as \(\Lambda\). The authors prove this for tiled orders \(\Lambda\).