Abstract: In this paper, we introduce the notions of $(m,n)$-coherent rings and $FP_{(m,n)}$-projective modules for nonnegative integers $m,n$. We prove that ($\mathcal{FP}_{(m,n)}$-Proj, ($\mathcal{FP}_{n}$-id)$_{\leq m}$) is a complete cotorsion pair for any $m,n\geq 0$ and it is hereditary if and only if the ring $R$ is a left $n$-coherent ring for all $m\geq 0$ and $n\geq 1$. Moreover, we study the existence of $\mathcal{FP}_{(m,n)}$-Proj covers and envelopes and obtain that if $\mathcal{FP}_{(m,n)}$-Proj is closed under pure quotients, then $\mathcal{FP}_{(m,n)}$-Proj is covering for any $n\geq2$. As applications, we obtain that every $R$-module has an epic $\mathcal{FP}_{(m,n)}$-Proj-envelope if and only if the left $FP_{(m,n)}$-global dimension of $R$ is at most 1 and $\mathcal{FP}_{(m,n)}$-Proj is closed under direct products.

Key words: $(m,n)$-coherent ring, $FP_{(m,n)}$-projective module, Cover , Envelope , Cotorsion pair