Strongly ( 𝒯 , n ) -coherent rings, ( 𝒯 , n ) -semihereditary rings and ( 𝒯 , n ) -regular rings

Zhanmin Zhu

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 3, page 657-674
  • ISSN: 0011-4642

Abstract

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Let 𝒯 be a weak torsion class of left R -modules and n a positive integer. A left R -module M is called ( 𝒯 , n ) -injective if Ext R n ( C , M ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a right R -module M is called ( 𝒯 , n ) -flat if Tor n R ( M , C ) = 0 for each ( 𝒯 , n + 1 ) -presented left R -module C ; a left R -module M is called ( 𝒯 , n ) -projective if Ext R n ( M , N ) = 0 for each ( 𝒯 , n ) -injective left R -module N ; the ring R is called strongly ( 𝒯 , n ) -coherent if whenever 0 → K → P → C → 0 is exact, where C is ( 𝒯 , n + 1 ) -presented and P is finitely generated projective, then K is ( 𝒯 , n ) -projective; the ring R is called ( 𝒯 , n ) -semihereditary if whenever 0 → K → P → C → 0 is exact, where C is ( 𝒯 , n + 1 ) -presented and P is finitely generated projective, then pd ( K ) ≤ n - 1 . Using the concepts of ( 𝒯 , n ) -injectivity and ( 𝒯 , n ) -flatness of modules, we present some characterizations of strongly ( 𝒯 , n ) -coherent rings, ( 𝒯 , n ) -semihereditary rings and ( 𝒯 , n ) -regular rings.

How to cite

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Zhu, Zhanmin. "Strongly $(\mathcal {T},n)$-coherent rings, $(\mathcal {T},n)$-semihereditary rings and $(\mathcal {T},n)$-regular rings." Czechoslovak Mathematical Journal 70.3 (2020): 657-674. <http://eudml.org/doc/297339>.

@article{Zhu2020,
abstract = {Let $\mathcal \{T\}$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(\mathcal \{T\},n)$-injective if $\{\rm Ext\}^n_R(C, M)=0$ for each $(\mathcal \{T\},n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(\mathcal \{T\},n)$-flat if $\{\rm Tor\}^R_n(M, C)=0$ for each $(\mathcal \{T\},n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(\mathcal \{T\},n)$-projective if $\{\rm Ext\}^n_R(M, N)=0$ for each $(\mathcal \{T\},n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(\mathcal \{T\},n)$-coherent if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal \{T\},n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(\mathcal \{T\},n)$-projective; the ring $R$ is called $(\mathcal \{T\},n)$-semihereditary if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal \{T\},n+1)$-presented and $P$ is finitely generated projective, then $\{\rm pd\} (K)\le n-1$. Using the concepts of $(\mathcal \{T\},n)$-injectivity and $(\mathcal \{T\},n)$-flatness of modules, we present some characterizations of strongly $(\mathcal \{T\},n)$-coherent rings, $(\mathcal \{T\},n)$-semihereditary rings and $(\mathcal \{T\},n)$-regular rings.},
author = {Zhu, Zhanmin},
journal = {Czechoslovak Mathematical Journal},
keywords = {$(\mathcal \{T\},n)$-injective module; $(\mathcal \{T\},n)$-flat module; strongly $(\mathcal \{T\},n)$-coherent ring; $(\mathcal \{T\},n)$-semihereditary ring; $(\mathcal \{T\},n)$-regular ring},
language = {eng},
number = {3},
pages = {657-674},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strongly $(\mathcal \{T\},n)$-coherent rings, $(\mathcal \{T\},n)$-semihereditary rings and $(\mathcal \{T\},n)$-regular rings},
url = {http://eudml.org/doc/297339},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Zhu, Zhanmin
TI - Strongly $(\mathcal {T},n)$-coherent rings, $(\mathcal {T},n)$-semihereditary rings and $(\mathcal {T},n)$-regular rings
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 657
EP - 674
AB - Let $\mathcal {T}$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(\mathcal {T},n)$-injective if ${\rm Ext}^n_R(C, M)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(\mathcal {T},n)$-flat if ${\rm Tor}^R_n(M, C)=0$ for each $(\mathcal {T},n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(\mathcal {T},n)$-projective if ${\rm Ext}^n_R(M, N)=0$ for each $(\mathcal {T},n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(\mathcal {T},n)$-coherent if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal {T},n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(\mathcal {T},n)$-projective; the ring $R$ is called $(\mathcal {T},n)$-semihereditary if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C$ is $(\mathcal {T},n+1)$-presented and $P$ is finitely generated projective, then ${\rm pd} (K)\le n-1$. Using the concepts of $(\mathcal {T},n)$-injectivity and $(\mathcal {T},n)$-flatness of modules, we present some characterizations of strongly $(\mathcal {T},n)$-coherent rings, $(\mathcal {T},n)$-semihereditary rings and $(\mathcal {T},n)$-regular rings.
LA - eng
KW - $(\mathcal {T},n)$-injective module; $(\mathcal {T},n)$-flat module; strongly $(\mathcal {T},n)$-coherent ring; $(\mathcal {T},n)$-semihereditary ring; $(\mathcal {T},n)$-regular ring
UR - http://eudml.org/doc/297339
ER -

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