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Torsion pairs over $$n$$-hereditary rings. (English) Zbl 1433.18005
Given a nonnegative integer $$n$$, one considers the class $$\mathcal{FP}_{n}$$ of all (left) modules $$M$$ over a ring $$R$$ such that there exists an exact sequence $F_{n} \rightarrow F_{n-1}\rightarrow \dots \rightarrow F_{0} \rightarrow M \rightarrow 0$ of $$R$$-modules, where $$F_{i}$$ is finitely generated and projective for all $$i \in \{ 0, \dots, n\}$$. This forms a decreasing sequence of subclasses of $$R$$-$$\operatorname{Mod}$$. Note that $$\mathcal{FP}_{0}$$ is the class of finitely generated $$R$$-modules.
In the article under review, the authors study homological properties associated to $$\mathcal{FP}_{n}$$, for every nonnegative integer $$n$$. More precisely, they introduce the notion of (left) $$n$$-hereditary ring (see Definition 3.1), which coincides with the usual notion of (left) hereditary ring if $$n = 0$$ and of (left) semi-hereditary ring if $$n = 1$$. One can also introduce the classes $$\mathcal{FP}_{n}$$-$$\operatorname{Inj}$$ and $$\mathcal{FP}_{n}$$-$$\operatorname{Flat}$$ of $$\operatorname{FP}_{n}$$-injective and $$\operatorname{FP}_{n}$$-flat $$R$$-modules, respectively, which coincide respectively with the classes of injective and flat modules if $$n=0$$. The authors then prove that the new notion of $$n$$-hereditary ring satisfies some of the usual expected homological properties, involving $$\operatorname{FP}_{n}$$-injective and $$\operatorname{FP}_{n}$$-flat $$R$$-modules (see Theorem 4.6). They also study the notion of $$n$$-coherent ring, which coincides with the usual notion of coherence if $$n = 1$$.
By their Theorem 4.6, the class $$\mathcal{FP}_{n}$$-$$\operatorname{Inj}$$ (resp., $$\mathcal{FP}_{n}$$-$$\operatorname{Flat}$$) is closed under direct sums, extensions and quotients (resp., direct sums, extensions and submodules), provided $$R$$ is $$n$$-hereditary. This allows them to consider the torsion pairs $$(\mathcal{FP}_{n}$$-$$\operatorname{Inj},\mathcal{FP}_{n}$$-$$\operatorname{Inj}^{\perp})$$ and $$({}^{\perp}\mathcal{FP}_{n}$$-$$\operatorname{Flat},\mathcal{FP}_{n}$$-$$\operatorname{Flat})$$ in that case (see Theorem 5.3).
The article includes several illustrative examples, and in particular they construct a Bézout ring, which is $$2$$-hereditary but not $$1$$-hereditary, for which the previous torsion pairs for $$n=2$$ are nontrivial (see Example 3.3, Proposition 5.6, and Propositions A.2–A.4).
##### MSC:
 1.8e+41 Torsion theories, radicals 1.6e+61 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
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