What does this research mean for the field?

Power series rings of the form A+XB[[X]] and A+XB[X] satisfy the ACC_d condition on ideals if and only if specific structural conditions are met, such as A being a semi-local PID with quotient field K or B being a finitely generated module over a Noetherian ring A. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to characterize power series rings that meet the ascending chain condition on ideals, linking this to properties of the underlying rings.

June 6, 2026

Power series rings with ACC d on ideals

Key Points

The aim is to characterize power series rings that meet the ascending chain condition on ideals, linking this to properties of the underlying rings.
Analyzed the relationship between power series rings and their subrings
Established conditions for the ascending chain condition on ideals based on ring properties like Noetherian and field extension
Proved implications of strong ACC on submodules and Noetherian conditions for quasi-local rings
Power series rings satisfy ACC on ideals if they are finite field extensions or semi-local PIDs.
Quasi-local subrings satisfy ACC if strong ACC on submodules holds, coupled with Noetherian conditions and integral elements.
For semi-local subrings in quasi-local rings of Krull dimension zero, ACC is met under similar Noetherian and finitely generated module conditions.

Abstract

A ring Formula: see text satisfies the divisibility condition on ascending chain of ideals (for short, ACC d on ideals) if, for every ascending chain Formula: see text of ideals of Formula: see text, there exists a positive integer Formula: see text such that for all Formula: see text, Formula: see text for some Formula: see text. Let Formula: see text be a subring of a field Formula: see text. We show that the ring Formula: see text (respectively Formula: see text) satisfies ACC d on ideals if and only if either Formula: see text is a field extension with finite degree or Formula: see text is a semi-local PID with quotient field Formula: see text. We prove that if Formula: see text is a quasi-local subring of a ring Formula: see text, then Formula: see text (respectively Formula: see text) satisfies ACC d on ideals if and only if Formula: see text satisfies strong ACC d on Formula: see text-submodules, Formula: see text is Noetherian and each non-unit of Formula: see text is integral over Formula: see text; if and only if either Formula: see text is Noetherian and Formula: see text is a finitely generated module over Formula: see text or Formula: see text is a rank one discrete valuation domain with quotient field Formula: see text. We prove that if Formula: see text is a semi-local subring of a quasi-local ring Formula: see text with Krull dimension zero, then Formula: see text (respectively Formula: see text) satisfies ACC d on ideals if and only if Formula: see text satisfies strong ACC d on Formula: see text-submodules and Formula: see text is Noetherian; if and only if either Formula: see text is Noetherian and Formula: see text is a finitely generated module over Formula: see text or Formula: see text is a semi-local PID with quotient field Formula: see text.

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Cite This Study

Mohamed Khalifa (Thu,) studied this question.

synapsesocial.com/papers/6a23ba3c71a5da9775e75f73 https://doi.org/https://doi.org/10.1142/s0219498827502549

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