  An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$
nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ 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, 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 N-reflexivity is weaker than that of reflexivity and stronger than
that of left N-right idempotent reflexivity and right idempotent reflexivity
which are introduced in Section 5.
