What question did this study set out to answer?

This research aims to establish the relationship between the minimaxness and Artinianness dimensions of finitely generated R-modules over commutative Noetherian rings.

June 14, 2026Open Access

On the Minimaxness and Artinianness Dimensions

Key Points

This research aims to establish the relationship between the minimaxness and Artinianness dimensions of finitely generated R-modules over commutative Noetherian rings.
Defined invariants AJI(M) and analyzed their properties under specific conditions.
Compared invariants across different ideals J and I within the context of finitely generated modules.
Applied the local-global principle to establish the equivalence of certain invariants.
Proved AJI(M) is equal to the infimum of conditions where JtHiI(M) remains Artinian and minimax across all t ∈ N0.
Demonstrated that invariants AII(M) and the infimum condition for 'not minimax' are equal.
Established the local-global principle, AJI(M) = inf{AJRpIRp(Mp)|p ∈ Spec (R)}, under certain scenarios.

Abstract

Let R be a commutative Noetherian ring, I, J be ideals of R such thatJ ⊆ I, and M be a finitely generated R-module. In this paper, we prove that theinvariants AJI (M): = infi ∈ N0 | JtHiI (M) is not Artinian for all t ∈ N0 and infi ∈N0 | JtHiI (M) is not minimax for all t ∈ N0 are equal. In particular, we show that theinvariants AII (M) and infi ∈ N0 | HiI (M) is not minimax are equal. We also establishthe local-global principle, AJI (M) = infAJRpIRp (Mp) |p ∈ Spec (R), in some cases.

Ask AI

Helpful

Bookmark

View Full Paper