Noetherianity from the ascending chain condition on ideals

Establish whether, for commutative ind-algebras in well-fibered pretannakian categories, the ascending chain condition on ideals implies that every submodule of $A\otimes X$ is finitely generated for every object $X$ of the category.

Background

The authors prove that, for commutative algebras in the relevant pretannakian categories, the descending chain condition on ideals is equivalent to being artinian. They observe that an analogous result connecting the ascending chain condition on ideals with Noetherianity would be natural.

Noetherianity is defined here by finite generation of every submodule of A⊗XA\otimes X for each object XX, and the authors explicitly leave the implication unresolved.

References

We will show that DCC on ideals is equivalent to being artinian for commutative algebras in $C$, see Theorem \ref{thm artin comm rings GRMN}, so it is natural to hope that in $C$ we also have that ACC on ideals implies being Noetherian, but we do not know if this holds.

— Absolutely flat algebras in tensor categories  (2608.13137 - Coulembier et al., 13 Aug 2026) in Remark in Section 5.3, “Artinian commutative algebras”