Symmetric radicals over Noetherian rings (Q5938589)

The object of study in this paper is a pair of radicals for a ring \(R\) -- that is, an ordered pair \((\sigma,\tau)\), where \(\sigma\) is a torsion radical defined on left \(R\)-modules, and \(\tau\) is a radical defined on right \(R\)-modules. Several variants of the notion of symmetry for such pairs are considered, where a pair is symmetric if \(\sigma(B)=\tau(B)\) for all \(R\)-\(R\)-bimodules \(B\) belonging to some class of \(R\)-\(R\)-bimodules. Symmetric pairs are characterised in the case that \(R\) is a Noetherian ring with finite classical Krull dimension, and a result of \textit{J. A. Beachy} [J. Pure Appl. Algebra 33, 243-252 (1984; Zbl 0547.16008)] proving the equivalence of various versions of symmetry for radical pairs is obtained under somewhat weaker hypotheses than before.