On the transfinite powers of the Jacobson radical of a DICC ring (Q2770395)

A ring is said to be DICC (double infinite chain condition) if no chain of right ideals exists that is both infinite ascending and infinite descending. The authors provide a counterexample to the assertion by \textit{O.~A.~S.~Karamzadeh} and \textit{M.~Motamedi} [Commun. Algebra 22, No. 6, 1933-1944 (1994; Zbl 0808.16024)] that some transfinite power of the Jacobson radical of such a ring is equal to zero. They also provide positive information on the transfinite powers of the Jacobson radical relative to a hereditary torsion theory, which specializes to give the correct information in the classical setting.