Artinian rings and modules everywhere

Abstract

For any commutative ring $A$, the finiteness conditions are a useful tool for approximating its structure. These finiteness conditions are reflected in some way in its spectrum; for example, if $A$ is a Noetherian ring, then Spec$(A)$ is a Noetherian topological space; the converse is not necessarily true. Noetherianness of Spec$(A)$ has an interesting consequence in the behaviour of hereditary torsion theories in Mod-$A$: they are of finite type; that is, for any hereditary torsion theory $\sigma$ in Mod-$A$ there exists a cofinal set of $\mathcal{L}(\sigma)$ consisting of finitely generated ideals. The aim of this work is to study rings and modules via finite type hereditary torsion theories. Therefore, we restrict ourselves to considering hereditary torsion theories defined by finitely generated ideals and finiteness conditions relative to these theories, extending some type of rings and modules as (totally) Noetherian, (totally) Artinian or Artinian$^*$.

Keywords

• Artinian ring
• Noetherian ring
• hereditary torsion theory
• finite type
• $S$–finite module

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