Modules and rings satisfying strong accr

Let Formula: see text be a commutative ring with identity and Formula: see text an Formula: see text-module. We say that Formula: see text satisfies strong accrFormula: see text if for every submodule Formula: see text of Formula: see text and for every sequence Formula: see text of elements of Formula: see text the ascending sequence of submodules of the form, Formula: see text is stationary. We say that a ring Formula: see text satisfies strong accrFormula: see text if Formula: see text regarded as a module over Formula: see text satisfies strong accrFormula: see text In this paper, we give a necessary and sufficient condition for the pulback (respectively, the Nagata’s idealization Formula: see text) to be strong accrFormula: see text ring. We also prove a new characterization for a valuation ring to be strong accrFormula: see text