On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings

Authors : Zhu ZHANMİN

Pages : 808-821

Doi:10.15672/hujms.624000

View : 13 | Download : 3

Publication Date : 2020-04-02

Article Type : Research Paper

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$^1_Rinsert ignore into journalissuearticles values(C, M);=0$ for every $C\in \mathscr{C}$; a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_Rinsert ignore into journalissuearticles values(M, E);=0$ 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.
Keywords : C coherent ring, strongly C coherent ring, C semihereditary ring