(m,n)-coherent Rings and FP(m,n)-projective Modules
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(FP(m,n)-Proj,(FPn-id)≤m)is a complete cotorsion pair for any m,n ≥ 0 and it is hereditary if and only if the ring R is a left n-coherent ring for all m ≥ 0 and n ≥ 1.Moreover,we study the existence of FP(m,n)-Proj covers and envelopes and obtain that if FP(m,n)-Proj is closed under pure quotients,then FP(m,n)-Proj is covering for any n ≥ 2.As applications,we obtain that every R-module has an epic FP(m,n)-Proj-envelope if and only if the left FP(m,n)-global dimension of R is at most 1 and FP(m,n)-Proj is closed under direct products.
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