An extension of S-artinian rings and modules to a hereditary torsion theory setting

Abstract

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