Local cohomology in classical rings (Q1802363)

Let \(R\) be a left noetherian ring. A torsion theory \(\sigma\) of \(R\)- modules is called symmetric if the filter of left ideals associated with \(\sigma\) has a cofinal subset of two-sided ideals. The properties of symmetric torsion theories are examined, and the stable symmetric torsion theories are characterized. \(R\) is called left classical if \(R\) is left noetherian and each symmetric torsion theory is stable. A new type of support for a module is introduced and studied for classical rings. A Mayer-Vietoris type of theorem is proved relative to symmetric stable torsion theories. For some symmetric torsion theories \(\sigma\) over a classical ring, relationships are given between: (1) chains of prime ideals, (2) the \(\sigma\)-dominant dimension, and (3) the vanishing of groups of local cohomology.