Cotorsion and tor pairs and finitistic dimensions over commutative rings. (English) Zbl 1436.13025
Droste, Manfred (ed.) et al., Groups, modules, and model theory – surveys and recent developments. In Memory of Rüdiger Göbel. Proceedings of the conference on new pathways between group theory and model theory, Mülheim an der Ruhr, Germany, February 1–4, 2016. Cham: Springer. 317-330 (2017).
Summary: Some of the most familiar cotorsion and Tor pairs on integral domains do not extend to rings \(R\) with divisors of zero. A closer look shows that for some of them the validity hinges on the strength of torsion-freeness in \(R\) which in turn is closely related to one of the finitistic projective and weak dimensions of the classical ring \(Q\) of quotients of \(R\). This interesting fact was observed by S. Bazzoni and D. Herbera; in their paper [Isr. J. Math. 174, 119–160 (2009; Zbl 1232.16009)] a number of important results of this kind can be found, explicitly or implicitly. We not only complement some of them with new results, but we also give different proofs for most of them, restricted to commutative rings. We consider three distinct versions of torsion-freeness, three finitistic dimensions of \(Q\), and investigate their influence on some cotorsion and Tor pairs of major interest.
For the entire collection see [Zbl 1372.20003].
For the entire collection see [Zbl 1372.20003].
MSC:
| 13C13 | Other special types of modules and ideals in commutative rings |
| 13C11 | Injective and flat modules and ideals in commutative rings |
Keywords:
cotorsion and Tor pairs; flat, torsion-free, torsion, and divisible modules; projective and weak dimension; finitistic dimension; injective; pure-injective; weak-injective modules; Matlis-, Enochs- and Warfield-cotorsion modulesCitations:
Zbl 1232.16009References:
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