In this paper we consider reduced (non-normal) commutative noetherian rings $R$. With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication with scalars in an integral extension of $R$. Using results of Greuel and Knorrer this yields a characterization of plane curves of finite Cohen--Macaulay type in terms of trace ideals. Further, we study one-dimensional local rings $(R,\mathfrak{m})$ such that that their normalization is isomorphic to the endomorphism ring $\mathrm{End}_R(\mathfrak{m})$: we give a criterion for this property in terms of the conductor ideal, and show that these rings are nearly Gorenstein. Moreover, using Grauert--Remmert normalization chains, we show the existence of noncommutative resolutions of singularities of low global dimensions for curve singularities.

中文翻译：

---

迹理想、归一化链和自同态环

在本文中，我们考虑约简（非正规）可交换诺特环 $R$。在某些$R$-模的导体理想和迹理想的帮助下，我们推导出自反$R$-模在与$R$的积分扩展中的标量相乘下闭合的准则。使用 Greuel 和 Knorrer 的结果，这产生了在迹理想方面的有限 Cohen-Macaulay 型平面曲线的特征。此外，我们研究了一维局部环 $(R,\mathfrak{m})$，使得它们的归一化与自同构环 $\mathrm{End}_R(\mathfrak{m})$ 同构：我们给出根据导体理想的这种性质的标准，并表明这些环接近 Gorenstein。此外，使用 Grauert--Remmert 归一化链，