Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings

IF 0.4 4区数学 Q4 MATHEMATICS

Algebra Colloquium Pub Date : 2022-07-26 DOI:10.1142/s1005386722000360

Dejun Wu, Hui-Shan Zhou

引用次数: 0

Abstract

Let [Formula: see text] and [Formula: see text] be rings and [Formula: see text] a [Formula: see text]-bimodule. If [Formula: see text] is flat and [Formula: see text] is finitely generated projective (resp., [Formula: see text] is finitely generated projective and [Formula: see text] is flat), then the characterizations of level modules and Gorenstein AC-projective modules (resp., absolutely clean modules and Gorenstein AC-injective modules) over the formal triangular matrix ring [Formula: see text] are given. As applications, it is proved that every Gorenstein AC-projective left [Formula: see text]-module is projective if and only if each Gorenstein AC-projective left [Formula: see text]-module and [Formula: see text]-module is projective, and every Gorenstein AC-injective left [Formula: see text]-module is injective if and only if each Gorenstein AC-injective left [Formula: see text]-module and [Formula: see text]-module is injective. Moreover, Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring [Formula: see text] are studied.

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形式三角矩阵环上的Gorenstein ac -射影模和ac -内射模

设[公式:见文]和[公式:见文]为环，[公式:见文]为双模。如果[Formula: see text]是平面的，而[Formula: see text]是有限生成的投影(见图1)。，[公式:见文]是有限生成的投影，[公式:见文]是平面的)，那么关卡模块和Gorenstein ac -投影模块的特征(见文)。给出了形式三角矩阵环上的绝对清洁模和Gorenstein ac -内射模[公式:见文]。作为应用，证明了当且仅当每个Gorenstein ac -射影左[公式:见文]-模都是射影，当且仅当每个Gorenstein ac -射影左[公式:见文]-模都是射影，并且每个Gorenstein ac -内射左[公式:见文]-模都是内射，当且仅当每个Gorenstein ac -内射左[公式:见文]-模都是内射。此外，研究了形式三角矩阵环上的Gorenstein ac -射影维数和ac -内射维数[公式:见文]。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algebra Colloquium 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

625

审稿时长

15.6 months

期刊介绍： Algebra Colloquium is an international mathematical journal founded at the beginning of 1994. It is edited by the Academy of Mathematics & Systems Science, Chinese Academy of Sciences, jointly with Suzhou University, and published quarterly in English in every March, June, September and December. Algebra Colloquium carries original research articles of high level in the field of pure and applied algebra. Papers from related areas which have applications to algebra are also considered for publication. This journal aims to reflect the latest developments in algebra and promote international academic exchanges.