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Separated sets of torsion theories. (English) Zbl 0695.16021
Several results on the frame R-tors of all (hereditary) torsion theories on R-mod are proved after various definitions have been given.
If $$\tau\in R$$-tors, then a nonzero left R-module N is $$\tau$$-cocritical if N is $$\tau$$-torsionfree but every proper homomorphic image of N is $$\tau$$-torsion and $$\tau$$ is good if for each torsion theory $$\sigma$$ on R-mod satisfying $$\tau <\sigma$$ there exists a $$\tau$$-cocritical $$\sigma$$-torsion left R-module. If $$\sigma <\tau$$ in R-tors, we say that the torsion theory $$\tau$$ is $$\sigma$$-uniform if the set of torsion theories $$\tau '$$ satisfying $$\sigma <\tau '\leq \tau$$ is closed under taking finite meets. A generalization of an element $$\sigma$$ in R-tors is an element $$\tau$$ of R-tors such that $$\sigma\leq \tau$$ and if U is a set of generalizations of $$\sigma$$ in R-tors, then U is $$\sigma$$-separated if $$\tau \wedge [\vee (U\setminus \{\tau \})]=\sigma$$ for each $$\tau$$ in U. If $$\sigma <\sigma '\leq \tau$$ in R-tors then $$\tau$$ is $$\sigma$$-essential over $$\sigma '$$ if and only if $$\sigma \neq \sigma '\wedge \sigma ''$$ for all $$\sigma <\sigma ''\leq \tau.$$
Some sample results are: (1) If $$\sigma$$ is a good torsion theory on R- mod then any two maximal $$\sigma$$-separated sets of $$\sigma$$-uniform torsion theories have the same cardinality. (2) If $$\sigma$$ is a good torsion theory on R-mod, then any proper generalization of $$\sigma$$ has a unique maximal $$\sigma$$-essential generalization.
Reviewer: F.Minnaar
MSC:
 16S90 Torsion theories; radicals on module categories (associative algebraic aspects)