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Journals / International Electronic Journal of Algebra / 2022 / Cilt: 32 - Sayı: 32

Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules

Mostafa AMINI
Driss BENNIS
Soumia MAMDOUHI
Pages
157–175
Publication Date
2022-07-16
DOI
—
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

Then we examine some of the conditions equivalent to that each right $R$-module is $n$-FCP-gr-projective.

📌 3 Key Takeaways

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Objective & Hypothesis:Let $R$ be a ring graded by a group $G$ and $n\geq1$ an integer.
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Key Findings & Data:We introduce the notion of $n$-FCP-gr-projective $R$-modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of $n$-FCP-gr-projective modules and graded right modules of $n$-FCP-gr-projective dimension at most $k$ over $n$-gr-cocoherent rings, (2) $R$ is right $n$-gr-cocoherent if and only if for every short exact sequence $0 \rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of graded right $R$-modules, where $B$ and $C$ are $n$-FCP-gr-projective, it follows that $A$ is $n$-FCP-gr-projective if and only if ($gr$-$\mathcal{FCP}_{n}$, $gr$-$\mathcal{FCP}_{n}^{\bot}$) is a hereditary cotorsion theory, where $gr$-$\mathcal{FCP}_n$ denotes the class of $n$-FCP-gr-projective right modules.
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Conclusion & Impact:Then we examine some of the conditions equivalent to that each right $R$-module is $n$-FCP-gr-projective.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:$n$-gr-Cocoherent ring · special gr-copresented module · $n$-FCP-gr-projective module

Abstract

Let $R$ be a ring graded by a group $G$ and $n\geq1$ an integer. We introduce the notion of $n$-FCP-gr-projective $R$-modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of $n$-FCP-gr-projective modules and graded right modules of $n$-FCP-gr-projective dimension at most $k$ over $n$-gr-cocoherent rings, (2) $R$ is right $n$-gr-cocoherent if and only if for every short exact sequence $0 \rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of graded right $R$-modules, where $B$ and $C$ are $n$-FCP-gr-projective, it follows that $A$ is $n$-FCP-gr-projective if and only if ($gr$-$\mathcal{FCP}_{n}$, $gr$-$\mathcal{FCP}_{n}^{\bot}$) is a hereditary cotorsion theory, where $gr$-$\mathcal{FCP}_n$ denotes the class of $n$-FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right $R$-module is $n$-FCP-gr-projective.

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Cite This Article

AMINI, M., BENNIS, D., MAMDOUHI, S. (2022). Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules. International Electronic Journal of Algebra, Cilt: 32 - Sayı: 32, 157–175. https://ajindex.com/articles/832197

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