Abstract
A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if for all [Formula: see text], [Formula: see text] is nilpotent, if and only if for all [Formula: see text], [Formula: see text] is strongly nil-clean, if and only if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is tripotent and [Formula: see text] is nil, if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] is a Boolean ring and [Formula: see text] is nil; [Formula: see text] is a Yaqub ring and [Formula: see text] is nil. Strongly 2-nil-clean group algebras are investigated as well.
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