$(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule
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| Format: | Preprint |
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2025
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| _version_ | 1866908632865570816 |
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| author | Amini, Mostafa Vahidi, Alireza Ghanavati, Fatemeh |
| author_facet | Amini, Mostafa Vahidi, Alireza Ghanavati, Fatemeh |
| contents | Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective (resp. $K_{d-1}$-$(n,d)$-flat) modules as a common generalization of some known modules such as $C$-injective, $C$-weak injective and $C$-$FP_n$-injective (resp. $C$-flat, $C$-weak flat and $C$-$FP_n$-flat) modules. Then we obtain some characterizations of two classes of these modules, namely $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$. We show that the cleasses $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$ are covering and preenveloping. Also, we investigate Foxby equivalence relative to the classes of this modules. Finally over $n$-coherent rings, we prove that the classes $\mathcal{I}_{ K_{d-1}}^{(n,d)}(R)_{<\infty}$ and $\mathcal{F}_{ K_{d-1}}^{(n,d)}(S)_{<\infty}$ are closed under extentions, kernels of epimorphisms and cokernels of monomorphisms.
Keywords: $K_{d-1}$-$(n,d)$-injective module; $K_{d-1}$-$(n,d)$-flat module; Foxby equivalence; special semidualizing bimodule. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04031 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule Amini, Mostafa Vahidi, Alireza Ghanavati, Fatemeh Rings and Algebras Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective (resp. $K_{d-1}$-$(n,d)$-flat) modules as a common generalization of some known modules such as $C$-injective, $C$-weak injective and $C$-$FP_n$-injective (resp. $C$-flat, $C$-weak flat and $C$-$FP_n$-flat) modules. Then we obtain some characterizations of two classes of these modules, namely $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$. We show that the cleasses $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$ are covering and preenveloping. Also, we investigate Foxby equivalence relative to the classes of this modules. Finally over $n$-coherent rings, we prove that the classes $\mathcal{I}_{ K_{d-1}}^{(n,d)}(R)_{<\infty}$ and $\mathcal{F}_{ K_{d-1}}^{(n,d)}(S)_{<\infty}$ are closed under extentions, kernels of epimorphisms and cokernels of monomorphisms. Keywords: $K_{d-1}$-$(n,d)$-injective module; $K_{d-1}$-$(n,d)$-flat module; Foxby equivalence; special semidualizing bimodule. |
| title | $(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2511.04031 |