THE DESCENDING CHAIN CONDITION RELATIVE TO A TORSION THEORY

A well-known theorem of Hopkins and Levitzki states that any left artinian ring with identity element is left noetherian. The main theorem of this paper generalizes this to the situation of a hereditary torsion theory with associated idempotent kernel functor σ. It is shown that if a ring R with identity element has the descending chain condition on σ-closed left ideals, then R has the ascending chain condition on σ-closed left ideals.