Abstract
Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.
Highlights
In this case, T is called a torsion class and its objects are called τ-torsion, F is called a torsion-free class, and its objects are called τ-torsion free
Proposition 2.1, Chap VI, in [9], a class T of right R-modules is a torsion class for some torsion theory if and only if T is closed under quotient modules, direct sums, and extensions
From Proposition 2.2, Chap VI in [9], a class F of right R-modules is a torsion-free class for some torsion theory if and only if F is closed under submodules, direct products, and extensions
Summary
T is called a torsion class and its objects are called τ-torsion, F is called a torsion-free class, and its objects are called τ-torsion free. R-module A is called τ-n-presented in case there exists an exact sequence of right R-modules 0 ⟶ Kn−1 ⟶ Fn−1 ⟶ · · · ⟶ F1 ⟶ F0 ⟶ A ⟶ 0, where each Fi is finitely generated free and Kn−1 is τ-finitely generated. Let A be a right R-module, τ a torsion theory of mod-R, and n a positive integer.
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