Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings
Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings
复制标题
迹理想与双有理扩张之间的对应关系及其在环 Gorenstein 性质分析中的应用
DOI：
10.1016/j.jpaa.2019.06.008
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发表时间：
2020
影响因子：
0.8
通讯作者：
Kumashiro Shinya
中科院分区：
文献类型：
--
作者：
Goto Shiro;Isobe Ryotaro;Kumashiro Shinya
研究了任意交换环上基环的迹理想集、稳定理想集和双有理扩张集之间的对应关系。如果基环是一维Gorenstein环，则对应是良好的。它表明，与一个极端的例外，满射的对应之一的特点Gorenstein性质的基环，只要它是一个Cohen-Macaulay局部环的维度。给出了交换Noether环上每个子模都是迹模的模的一个刻划。引入了反稳定环的概念，探讨了反稳定环的基本性质。
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties.