A ring R is called NLI (rings whose nilpotent elements form a Lie ideal) if for each a ∈ N(R) and b ∈ R, ab − ba ∈ N(R). Clearly, NI rings are NLI. In this note, many properties of NLI rings are studied. The main results we obtain are the following: (1) NLI rings are directly finite and left min-abel; (2) If R is a NLI ring, then (a) R is a strongly regular ring if and only if R is a Von Neumann regular ring; (b) R is (weakly) exchange if and only if R is (weakly) clean; (c) R is a reduced ring if and only if R is a n-regular ring; (3) If R is a NLI left MC2 ring whose singular simple left modules are Wnil-injective, then R is reduced.
Badawi, A., On abelian II-regular rings, Comm. Algebra, 25(4) (1997), 1009–1021.
Badawi A., 'On abelian II-regular rings' (1997) 25Comm. Algebra: 1009-1021.