Trace ideals, normalization chains, and endomorphism rings (Q2214900)

Let \(R\) be a reduced commutative Noetherian ring. Let \(M\) be a finitely generated reflexive \(R\)-module and \(R'\) a finite birational extension of \(R.\) It is shown that \(M\) is a \(R'\)-module if and only if \(\tau(M)\subset\mathcal{C}_{R'/R}\), where \(\tau(M)\) is the trace ideal of \(M\) and \(\mathcal{C}_{R'/R}\) is the conductor of \(R'\) in \(R.\) As a corollary it is shown that a reduced local complete ring of dimension \(1\) and embedding dimension \(2\) containing \(\mathbb{Q}\) is of finite CM-type if and only if there are finitely many possibilities for \(\tau(M),\) where \(M\) is a CM-moule over \(R\). Then one-dimensional local rings \((R,\mathfrak{m})\) such that their normalization is \(\mathrm{End}_R(\mathfrak{m})\) are studied. A criterion for this property in terms of the conductor ideal is given and it is shown that these rings are nearly Gorenstein.