** Authors:**
LONG WANG, JUNCHAO WEI, LIBIN LI

** Abstract: **
A ring R is called NZI if for any a \in R, l(a) is an N-ideal of
R. In this paper, we first study some basic properties and basic
extensions of NZI rings. Next, we study the strong regularity of NZI
rings and obtain the following results: (1) Let R be a left SF-ring.
Then R is a strongly regular ring if and only if R is an NZI ring;
(2) If R is an NZI left MC2 ring and every simple singular left
R-module is nil-injective, then R is reduced; (3) Let R be an
NZI ring. Then R is a strongly regular ring if and only if R is a
von Neumann regular ring; (4) Let R be an NZI ring. Then R is a
clean ring if and only if R is an exchange ring.

** Keywords: **
Reduced ring, SF ring, strongly regular ring, N-ideal, NZI
ring, nil-injective

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