Some remarks about $FP_{n}$-projective and $FP_{n}$-injective modules

TL;DR

This paper explores new characterizations of $FP_n$-projective and $FP_n$-injective modules, extends known results, and introduces a global dimension measure for rings related to their Noetherian property.

Contribution

It provides novel characterizations of $FP_n$-modules, extends existing results, and defines a new global dimension to assess how non-Noetherian a ring is.

Findings

New characterizations of $FP_n$-projective modules

Extended results on $FP_n$-injective global dimension

Introduced a global dimension measuring non-Noetherian nature

Abstract

Let $R$ be a ring. In \cite{MD4} Mao and Ding defined an special class of $R$-modules that they called \( FP_n \)-projective $R$-modules. In this paper, we give some new characterizations of \( FP_n \)-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the \( FP_n \)-injective global dimension in terms of \( FP_n \)-projective $R$-modules are obtained. Using the \( FP_n \)-projective dimension of an $R$-module defined by Ouyang, Duan and Li in \cite{Ouy} we introduce a slightly different \( FP_n \)-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of \cite{Ouy} when the ring in question is strong $n$-coherent.