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