An extension of s-noetherian rings and modules

Abstract

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 totally noetherian w.r.t. $\sigma$ is a local property, and if $A$ is a totally noetherian ring w.r.t $\sigma$, then $\sigma$ is of fi nite type.