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 if $A$ is a totally $\sigma$-noetherian ring, then $\sigma$ is of finite type, and that totally $\sigma$-noetherian is a local property.