Reflexive ideals and reflexively closed subsets in rings

Abstract

We continue the study of the reflexivity of ideals, introduced by Mason, and extend this notion to the subsets in rings. We first construct the smallest reflexive ideal containing [Formula: see text] from any proper ideal [Formula: see text] of any given ring [Formula: see text]; by which we can construct reflexive ideals but not semiprime in a kind of noncommutative ring. A subset [Formula: see text] of a ring [Formula: see text] is called reflexively closed if [Formula: see text] for [Formula: see text] implies [Formula: see text], checking that a ring [Formula: see text] is symmetric if and only if the right (left) annihilator of [Formula: see text] is reflexively closed for any [Formula: see text]. We prove that the set of all nilpotent elements in a ring [Formula: see text] is reflexively closed if and only if [Formula: see text] is nil for any nilpotent element [Formula: see text] in [Formula: see text]; and that the Köthe’s conjecture holds if and only if the union (sum) of the upper nilradical and any nil right ideal is reflexively closed. We provide another process to show that the set of all nilpotent elements of the polynomial ring over an NI ring need not be reflexively closed.