Abstract
A regular ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A≅ A⊕ B≅ A⊕ B implies that A≅ B. We prove, in this article, that a regular ring R in which 2 is invertible is separative if and only if each a ∈ R satisfying R(1 − a 2)R = Rr(a) = ℓ(a)R and i(End R (aR)) = ∞ is unit-regular if and only if each a ∈ R satisfying R(1 − a 2)R ∩ RaR = Rr(a) ∩ ℓ(a)R ∩ RaR and i(End R (aR)) = ∞ is unit-regular. Further equivalent characterizations of such regular rings are also obtained.
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Communicated by V. A. Artamonov.