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Finiteness conditions and cotorsion pairs. (English) Zbl 1362.18019
The authors consider the classes of finitely $$n$$-presented modules $$\mathcal{FP}_n$$ and $$n$$-coherent rings and establish a characterization of $$n$$-coherent rings in terms of finitely $$n$$-presented modules.
Then, the relative homological algebra with respect to the class $$\mathcal{FP}_n$$, from the injective and flat properties is studied to get the classes of modules $$\mathcal{FP}_n$$-Inj and $$\mathcal{FP}_n$$-Flat, respectively. Some of the presented results on this matter are adopted from D. Bravo, J. Gillespie and M. Hovey [“The stable module category of a general ring”, Preprint, arXiv:1405.5768] and J. Chen and N. Ding [Commun. Algebra 24, No. 10, 3211–3216 (1996; Zbl 0877.16010)].
At the end, the completeness of certain cotorsion pairs associated to the classes $$\mathcal{FP}_n$$-Inj and $$\mathcal{FP}_n$$-Flat is investigated.

MSC:
 18G25 Relative homological algebra, projective classes (category-theoretic aspects) 16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
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References:
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