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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
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)