Reflexivity of rings via nilpotent elements

Abstract

An ideal I of a ring R is called left N-reflexive if for any a ∈ nil(R) and b ∈ R, aRb ⊆ I implies bRa ⊆ I, where nil(R) is the set of all nilpotent elements of R. The ring R is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive rings, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal I of a ring R, R/I is left N-reflexive. If an ideal I of a ring R is reduced as a ring without identity and R/I is left N-reflexive, then R is left N-reflexive. If R is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in R[x] are nilpotent in R, it is proved that R is left N-reflexive if and only if R[x] is left N-reflexive. We show that the concept of left N-reflexivity is weaker than that of reflexivity and stronger than that of right idempotent reflexivity.