  For any commutative ring $A$ we introduce a generalization of $S$--artinian
rings using a hereditary torsion theory $\sigma$ instead of a multiplicative
closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally
$\sigma$--artinian ring, then $\sigma$ must be of finite type, and $A$ is
totally $\sigma$--noetherian.
