Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| author | Amini, Mostafa Vahidi, Alireza Rezaei, Farideh |
| author_facet | Amini, Mostafa Vahidi, Alireza Rezaei, Farideh |
| contents | Let $R$ and $S$ be rings, $C= {}_SC_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\infty$. In this paper, we introduce the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules as a common generalization of some known modules such as $C$-$FP_{n}$-injective (resp. $C$-weak injective) $R$-modules and $C$-$FP_{n}$-flat (resp. $C$-weak flat) $S$-modules. Then we investigate $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions of modules, where the classes of these modules, namely $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$, respectively. We study Foxby equivalence relative to these classes, and also the existence of $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$ preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02277 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules Amini, Mostafa Vahidi, Alireza Rezaei, Farideh Rings and Algebras Commutative Algebra Let $R$ and $S$ be rings, $C= {}_SC_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\infty$. In this paper, we introduce the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules as a common generalization of some known modules such as $C$-$FP_{n}$-injective (resp. $C$-weak injective) $R$-modules and $C$-$FP_{n}$-flat (resp. $C$-weak flat) $S$-modules. Then we investigate $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions of modules, where the classes of these modules, namely $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$, respectively. We study Foxby equivalence relative to these classes, and also the existence of $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$ preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings. |
| title | Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules |
| topic | Rings and Algebras Commutative Algebra |
| url | https://arxiv.org/abs/2210.02277 |