Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Goto, Shiro¹  Isobe, Ryotaro²  Kumashiro, Shinya² 

[1] Meiji Univ, Sch Sci & Technol, Dept Math, Tama Ku, 1-1-1 Higashi Mita, Kawasaki, Kanagawa 2148571, Japan

[2] Chiba Univ, Grad Sch Sci & Engn, Dept Math & Informat, Inage Ku, Yayoi Cho 1-33, Chiba 2638522, Japan

关键词: Cohen-Macaulay ring;    Gorenstein ring;    Trace module;    Trace ideal;    Stable ideal;    Stable ring;   

DOI  :  10.1016/j.jpaa.2019.06.008

来源: Elsevier

【 摘 要 】

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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free