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.

💡 Research Summary

The paper investigates the homological classes of FPₙ‑projective and FPₙ‑injective modules over an arbitrary associative ring R, extending the classical theory of FP‑projective (absolutely pure) modules. After recalling the definition of finitely n‑presented modules, the authors introduce FPₙ‑projective modules as those P satisfying Ext¹_R(P,M)=0 for every FPₙ‑injective module M. They prove several new characterizations: a finitely generated module is FPₙ‑projective if and only if it is finitely n‑presented; equivalently, a module P is FPₙ‑projective precisely when every short exact sequence 0→A→B→P→0 with A in FPₙ‑Inj splits. This reformulation replaces the Ext‑condition with a splitting condition and shows that FPₙ‑projective modules are exactly the direct summands of S‑filtered modules, where S is a set of representatives of all finitely n‑presented modules.

The authors then study n‑pure exact sequences, a generalization of pure exact sequences, defined by the property that every finitely n‑presented module is projective with respect to the sequence. Using a recent result of Tan, Wang and Zhao, they prove that an exact sequence 0→A→B→C→0 is n‑pure iff the induced sequence 0→M⊗_R A→M⊗_R B→M⊗_R C→0 is exact for every finitely n‑presented right R‑module M. This leads to a clean characterization of FPₙ‑flat modules: a module C is FPₙ‑flat precisely when every short exact sequence ending in C is n‑pure. Consequently, the class of FPₙ‑flat modules coincides with the class of modules that turn all exact sequences into n‑pure ones.

Building on the work of Ouyang, Duan and Li, the paper introduces the FPₙ‑projective dimension of a module and defines a new global invariant, the FPₙ‑projective global dimension of R, denoted FPₙ‑gldim(R). This invariant measures how far R is from being Noetherian: when R is strong n‑coherent, FPₙ‑gldim(R) agrees with the (n,0)‑projective global dimension of Ouyang et al.; when R is Noetherian, it coincides with the classical global dimension. The authors compare this new dimension with the λ‑dimension and the usual global dimension, establishing inequalities and equalities under various coherence assumptions.

A significant portion of the paper is devoted to the behavior of FPₙ‑projective modules under submodules. The authors define FP‑hereditary rings as those for which the class FPₙ‑Proj is closed under submodules for every n. They prove that a ring is FP‑hereditary if and only if it is strong n‑coherent and the FPₙ‑projective global dimension is zero. This yields a generalization of the classical hereditary ring theory to the FPₙ‑setting.

In later sections the authors compute the FPₙ‑injective global dimension using FPₙ‑projective modules. They show that a ring is an (n,d)‑ring (i.e., every FPₙ‑injective module has injective dimension ≤ d) precisely when the FPₙ‑projective global dimension is ≤ d. As corollaries they obtain new characterizations of n‑von Neumann regular rings (all finitely n‑presented modules are projective) and n‑hereditary rings (FPₙ‑projective modules are closed under kernels of epimorphisms).

The final section outlines applications to subprojectivity domains, the C‑F conjecture, and trace modules in FPₙ‑injective envelopes, illustrating how the developed theory can be employed in concrete module‑theoretic problems.

Overall, the paper provides a comprehensive extension of FP‑projective/injective theory to the finitely n‑presented context, introduces a novel global homological invariant, and connects these concepts with strong n‑coherence, purity, and classical dimensions, thereby opening new avenues for research in relative homological algebra and non‑commutative ring theory.