形式三角矩阵环上的Gorenstein ac -射影模和ac -内射模

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

Dejun Wu, Hui-Shan Zhou

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引用次数: 0

摘要

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]。

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

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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