On $\phi$-$w$-Flat modules and Their Homological Dimensions

Xiaolei Zhang; Wei Zhao

On $\phi$-$w$-Flat modules and Their Homological Dimensions

Commutative Algebra 2023-02-23 v2

Authors: Xiaolei Zhang , Wei Zhao

Abstract

In this paper, we introduce and study the class of $\phi$ - $w$ -flat modules which are generalizations of both $\phi$ -flat modules and $w$ -flat modules. The $\phi$ - $w$ -weak global dimension $\phi$ - $w$ -w.gl.dim $(R)$ of a strongly $\phi$ -ring $R$ is also introduced and studied. We show that, for a strongly $\phi$ -ring $R$ , $\phi$ - $w$ -w.gl.dim $(R)=0$ if and only if $w$ - $dim(R)=0$ if and only if $R$ is a $\phi$ -von Neumann ring. It is also proved that, for a strongly $\phi$ -ring $R$ , $\phi$ - $w$ -w.gl.dim $(R)\leq 1$ if and only if each nonnil ideal of $R$ is $\phi$ - $w$ -flat, if and only if $R$ is a $\phi$ -PvMR, if and only if $R$ is a PvMR.

Keywords

module theory commutative algebra

Cite

@article{arxiv.2107.12643,
  title  = {On $\phi$-$w$-Flat modules and Their Homological Dimensions},
  author = {Xiaolei Zhang and Wei Zhao},
  journal= {arXiv preprint arXiv:2107.12643},
  year   = {2023}
}

Comments

Revise the published version. Note in this version, when we study $\phi$-$w$-weak global dimension, we always require the ring is a a strongly $\phi$-ring. arXiv admin note: text overlap with arXiv:2107.11907

On $\phi$-$w$-Flat modules and Their Homological Dimensions

Abstract

Keywords

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