Volume 21, Issue 2 (9-2026)                   IJMSI 2026, 21(2): 61-78 | Back to browse issues page

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Amini M. Gorenstein $(n‎, ‎d)$-Injective and‎ ‎$(n‎, ‎d)$-flat Modules. IJMSI 2026; 21 (2) :61-78
URL: http://ijmsi.ir/article-1-2340-en.html
Gorenstein $(n‎, ‎d)$-Injective and‎ ‎$(n‎, ‎d)$-flat Modules
Mostafa Amini
Abstract:  
‎Let $n‎, ‎d$ be non-negative integers‎. ‎We introduce and study the notions of Gorenstein $(n,d)$-injective and Gorenstein $(n,d)$-flat modules by using the notion of special finitely presented‎, ‎$(n,d)$-injective and $(n,d)$-flat modules for any $ngeq d+1$‎. ‎Then we investigate the properties of these modules in case that every special finitely presented module with finite projective dimension (resp‎. ‎flat dimension) is $2$-presented‎. ‎Moreover‎, ‎as applications over special coherent rings‎, ‎we investigate the relationships between Gorenstein $(n,d)$-injective modules and Gorenstein $(n,d)$-flat modules‎, ‎then we‎ ‎obtain that the classes ${mathcal{GI}_d^n(R)}$ and ${mathcal{GF}_d^n(R^{op})}$ are covering and preenveloping‎, ‎where ${mathcal{GI}_d^n(R)}$ and ${mathcal{GF}_d^n(R^{op})}$ denote the subcategories of Gorenstein $(n,d)$-injective left $R$-module and Gorenstein $(n,d)$-flat right $R$-modules‎, ‎respectively.
Keywords: $(n, d)$-Flat module, $(n, d)$-Injective module, Gorenstein $(n, d)$-flat module, Gorenstein $(n, d)$-injective module, Special finitely presented module.
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Type of Study: Research paper | Subject: General
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