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

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.