** Authors:**
JUNCHAO WEI, LIBIN LI

** Abstract: **
In this paper, we show the following results: (1) R is a
min-leftsemicentral ring if and only if eR(1-e)Re=0 for all e \in
ME_l(R); (2) Quasi-normal rings, NI rings and weakly reversible rings
are all min-leftsemicentral ring; (3) R is left MC2 ring if and only
if aRe=0 implies eRa=0 for all e \in ME_l(R) and a \in R if and only
if every projective simple left R-module is MUP-injective; (4) R is
reduced if and only if R is n-regular and quasi-normal if and only if
R is n-regular and weakly reversible; (5) R is strongly regular if and
only if R is regular and quasi-normal if and only if R is regular and
weakly reversible.

** Keywords: **
Min-leftsemicentral rings, quasi-normal rings. NCI
rings, weakly reversible rings, left MC2 rings, directly finite
rings, regular rings

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