Reflexive property on nil ideals

Abstract

We in this note consider the reflexive ring property on nil ideals, introducing the concept of a nil-reflexive ring as a generalization of the reflexive ring property. We will call a ring [Formula: see text] nil-reflexive if [Formula: see text] implies [Formula: see text] for nil ideals [Formula: see text] of [Formula: see text]. The polynomial and the power series rings over a right Noetherian ring (or an NI ring) [Formula: see text] are shown to be nil-reflexive if [Formula: see text] implies [Formula: see text] for all [Formula: see text]. We further investigate the structure of nil-reflexive rings, related to various sorts of ring extensions which have roles in ring theory.