An extension of $S$--noetherian rings and modules

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