In this paper, we give some new characterizations and properties of left $\mathscr{C}$-coherent rings, left strongly $\mathscr{C}$-coherent rings and left $\mathscr{C}$-semihereditary rings.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $R$ be a ring and $\mathscr{C}$ be a class of some finitely presented left $R$-modules.
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Key Findings & Data:A left $R$-module $M$ is called $\mathscr{C}$-injective if Ext$^1_R(C, M)=0$ for every $C\in \mathscr{C}$; a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_R(M, E)=0$ for any $\mathscr{C}$-injective module $E$. $R$ is called left $\mathscr{C}$-coherent if every $C\in \mathscr{C}$ is 2-presented; $R$ is called left strongly $\mathscr{C}$-coherent, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ and $P$ is finitely generated projective, then $K$ is $\mathscr{C}$-projective; a ring $R$ is called left $\mathscr{C}$-semihereditary, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ , $P$ is finitely generated projective, then $K$ is projective.
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Conclusion & Impact:In this paper, we give some new characterizations and properties of left $\mathscr{C}$-coherent rings, left strongly $\mathscr{C}$-coherent rings and left $\mathscr{C}$-semihereditary rings.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:C -coherent ring · strongly C -coherent ring · C -semihereditary ring
Abstract
Let $R$ be a ring and $\mathscr{C}$ be a class of some finitely presented left $R$-modules. A left $R$-module $M$ is called $\mathscr{C}$-injective if Ext$^1_R(C, M)=0$ for every $C\in \mathscr{C}$; a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_R(M, E)=0$ for any $\mathscr{C}$-injective module $E$. $R$ is called left $\mathscr{C}$-coherent if every $C\in \mathscr{C}$ is 2-presented; $R$ is called left strongly $\mathscr{C}$-coherent, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ and $P$ is finitely generated projective, then $K$ is $\mathscr{C}$-projective; a ring $R$ is called left $\mathscr{C}$-semihereditary, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ , $P$ is finitely generated projective, then $K$ is projective. In this paper, we give some new characterizations and properties of left $\mathscr{C}$-coherent rings, left strongly $\mathscr{C}$-coherent rings and left $\mathscr{C}$-semihereditary rings.
Zhanmin, Z. (2022). On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 2, 808–821. https://ajindex.com/articles/529608
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