Modules with n-acc and the acc on certain types of annihilators

A unitary module M over a commutative ring R with unity satisfies “acc on d-colons,” if for every submodule N of M and every sequence (an)n of elements of R the ascending chain N:a1 N:a1a2 stabilizes. In this paper we study the acc on d-colons and show that it implies the acc on n-generated submodules for every n (“pan-acc”), generalizing a result of W. Heinzer and D. Lantz. The method involves a “generalized Nakayamaʹs Lemma” for these modules. Further it is shown that every R-module RI for a noetherian ring R satisfies pan-acc, and we give a sufficient condition for the acc on principal ideals to rise to the polynomial ring.