Cartan-Eilenberg projective, injective and flat complexes

Xiaorui Zhai; Chunxia Zhang

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 151-167
  • ISSN: 0011-4642

Abstract

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Let R be an associative ring with identity and ℱ a class of R -modules. In this article: we first give a detailed treatment of Cartan-Eilenberg ℱ complexes and extend the basic properties of the class ℱ to the class CE ( ℱ ). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.

How to cite

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Zhai, Xiaorui, and Zhang, Chunxia. "Cartan-Eilenberg projective, injective and flat complexes." Czechoslovak Mathematical Journal 66.1 (2016): 151-167. <http://eudml.org/doc/276809>.

@article{Zhai2016,
abstract = {Let $R$ be an associative ring with identity and $\mathcal \{F\}$ a class of $R$-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg $\mathcal \{F\}$ complexes and extend the basic properties of the class $\mathcal \{F\}$ to the class $\{\rm CE\}(\mathcal \{F\}$). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, $\rm QF$ rings, semisimple rings, hereditary rings and perfect rings.},
author = {Zhai, Xiaorui, Zhang, Chunxia},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cartan-Eilenberg projective complex; Cartan-Eilenberg injective complex; Cartan-Eilenberg flat complex},
language = {eng},
number = {1},
pages = {151-167},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cartan-Eilenberg projective, injective and flat complexes},
url = {http://eudml.org/doc/276809},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Zhai, Xiaorui
AU - Zhang, Chunxia
TI - Cartan-Eilenberg projective, injective and flat complexes
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 151
EP - 167
AB - Let $R$ be an associative ring with identity and $\mathcal {F}$ a class of $R$-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg $\mathcal {F}$ complexes and extend the basic properties of the class $\mathcal {F}$ to the class ${\rm CE}(\mathcal {F}$). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, $\rm QF$ rings, semisimple rings, hereditary rings and perfect rings.
LA - eng
KW - Cartan-Eilenberg projective complex; Cartan-Eilenberg injective complex; Cartan-Eilenberg flat complex
UR - http://eudml.org/doc/276809
ER -

References

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