Quasi-reflexive rings with non-nilpotent minimal one-sided ideals

In this paper we show that a ring is a member of the class of rings named in the title if and only if the ring is quasi-reflexive and contains at least one idempotent canonical quasi-ideal.We also prove the latter criterion is equivalent to several other ones.To attain that, we introduce the concept of a left n-socle and dually that of a right n-socle for arbitrary rings.An example is displayed to show that the presence of e.g.a nonzero right n-socle in a ring does not ensure the existence of a nonzero left n-socle.But in the quasi-reflexive case, it turns out that the notion of a left n-socle coincides with the right one.Finally, we give decomposition results which mainly deal with nonzero n-socles of quasi-reflexive rings and their semigroups n-socles concordantly, thereby, generalizing corresponding work in the semiprime case.