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 leftn-socle and dually that of a rightn-socle for arbitrary rings.An example is displayed to show that the presence of e.g.a nonzero rightn-socle in a ring does not ensure the existence of a nonzero leftn-socle.But in the quasi-reflexive case, it turns out that the notion of a leftn-socle coincides with the right one.Finally, we give decomposition results which mainly deal with nonzeron-socles of quasi-reflexive rings and their semigroupsn-socles concordantly, thereby, generalizing corresponding work in the semiprime case.