Critical and injective modules over skew polynomial rings

被引：1

作者：

Brown, Ken ^{[1
]}

Carvalho, Paula A. A. B. ^{[2
]}

Matczuk, Jerzy ^{[3
]}

机构：

[1] Univ Glasgow, Sch Math & Stat, Glasgow G12 8SQ, Scotland

[2] Univ Porto, Fac Ciencias, Dept Matemat, CMUP, Rua Campo Alegre S-N, P-4169007 Porto, Portugal

[3] Univ Warsaw, Inst Math, Banacha 2, PL-02097 Warsaw, Poland

来源：

JOURNAL OF PURE AND APPLIED ALGEBRA | 2023年 / 227卷 / 11期

关键词：

Injective module; Noetherian ring; Simple module; Skew polynomial ring; EXTENSIONS;

D O I：

10.1016/j.jpaa.2023.107441

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let & alpha; be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[& theta;; & alpha;]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian -that is, for S to satisfy property (O). It is easy and well known that if & alpha; is of finite order, then S has this property, but in order to get the criterion when & alpha; has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that S ⠂ = k[[X]][& theta;, & alpha;] satisfies (o) for all k-algebra automorphisms & alpha; of k[[X]].& COPY; 2023 Elsevier B.V. All rights reserved.

引用

页数：26

共 12 条

[1] The simple modules of certain generalized crossed products [J].