Generalized Gorenstein Modules

IF 0.4 4区 数学 Q4 MATHEMATICS Algebra Colloquium Pub Date : 2022-12-01 DOI:10.1142/s1005386722000463
A. Iacob
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引用次数: 3

Abstract

We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.
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广义Gorenstein模
我们引入了Gorenstein内射模的一种推广:Gorenstein[公式:见文]-内射模(用[公式:见文]表示)。它们是内射模的精确复合体的循环,当我们将函子[公式:见文]应用于[公式:见文]任何[公式:见文]内射模时,它们仍然是精确的。因此,[公式:见文]为经典Gorenstein内射模类,[公式:见文]为Ding内射模类。我们证明了在任意环上,对于任意[公式:见文],类[公式:见文]是完美扭转对的右半部分,因此它是一个包络类。对于[公式:见文],我们证明[公式:见文](即,丁注射剂)构成遗传扭转对的右半部分。此外,如果环[公式:见文本]是相干的,则丁内射模形成一个包络类。我们还定义了对偶概念,即Gorenstein的[公式:见文]-投影(用[公式:见文]表示)。它们推广了Ding投影模,也推广了Gorenstein投影模。我们证明了对于任何[公式:见文]类[公式:见文]是完全遗传扭转对的左半部分,因此它是特殊覆盖。
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
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.
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