Power series rings with ACC d on ideals

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.