A module M M over a ring R R is said to satisfy (accr) if the ascending chain of residuals of the form N : B ⊆ N : B 2 ⊆ N : B 3 ⊆ ⋯ N: B \subseteq N:{B^2} \subseteq N:{B^3} \subseteq \cdots terminates for every submodule N N and every finitely generated ideal B B of R R . A ring satisfies (accr) if it does as a module over itself. This class of rings and modules satisfies various properties of Noetherian rings and modules. For each of the following rings, we investigate a necessary and sufficient condition for the ring to satisfy (accr): polynomial rings, power series rings, valuation rings, and Prüfer domains. We also prove that if R R is a ring satisfying (accr), then every finitely generated R R -module satisfies (accr).