Modules and rings satisfying strong accr

Abstract

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 accr[Formula: 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 accr[Formula: see text] if [Formula: see text] regarded as a module over [Formula: see text] satisfies strong accr[Formula: 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 accr[Formula: see text] ring. We also prove a new characterization for a valuation ring to be strong accr[Formula: see text]