Equality of semiprime and prime radicals for left ideals in general rings

Determine whether, for every associative unital ring R and every left ideal I ⊆ R, the semiprime radical of I (the smallest semiprime left ideal containing I) equals the prime radical of I (the intersection of all prime left ideals of R that contain I). Equivalently, ascertain whether the equality of these two one-sided radicals holds for arbitrary rings, beyond the classes covered by sufficient conditions established in the paper.

Background

The paper introduces one-sided analogues of radicals for left ideals: the semiprime radical of a left ideal I is defined as the smallest semiprime left ideal containing I, while the prime radical is defined as the intersection of all prime left ideals containing I. Since every prime left ideal is semiprime, the prime radical is always contained in the Jacobson radical of I.

Rings in which every semiprime left ideal equals the intersection of all prime left ideals that contain it are termed left intersection rings. The authors provide sufficient conditions for equality of these radicals (e.g., for rings finitely generated as modules over their centers and for Azumaya algebras), but the general case remains unresolved.