Volume 29, Issue 1
$\mathcal{F}$-Perfect Rings and Modules

Bo Lu

Commun. Math. Res., 29 (2013), pp. 41-50.

Published online: 2021-05

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  • Abstract

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.

  • Keywords

$\mathcal{F}$-Perfect ring, $\mathcal{F}$-cover, $\mathcal{F}$-perfect module, cotorsion theory, projective module.

  • AMS Subject Headings

16D50, 16D40, 16L30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-29-41, author = {Lu , Bo}, title = {$\mathcal{F}$-Perfect Rings and Modules}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {29}, number = {1}, pages = {41--50}, abstract = {

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19027.html} }
TY - JOUR T1 - $\mathcal{F}$-Perfect Rings and Modules AU - Lu , Bo JO - Communications in Mathematical Research VL - 1 SP - 41 EP - 50 PY - 2021 DA - 2021/05 SN - 29 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19027.html KW - $\mathcal{F}$-Perfect ring, $\mathcal{F}$-cover, $\mathcal{F}$-perfect module, cotorsion theory, projective module. AB -

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.

Bo Lu. (2021). $\mathcal{F}$-Perfect Rings and Modules. Communications in Mathematical Research . 29 (1). 41-50. doi:
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