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.
An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring R is uniquely π-clean if some power of every element in R is uniquely clean. In this article, we prove that a ring R is uniquely π-clean if and only if for any a ∈ R, there exists an m ∈ N and a central idempotent e ∈ R such that a m − e ∈ J(R), if and only if R is abelian; every idempotent lifts modulo J(R); and R/P is torsion for all prime ideals P containing the Jacobson radical J(R). Further, we prove that a ring R is uniquely π-clean and J(R) is nil if and only if R is an abelian periodic ring, if and only if for any a ∈ R, there exists some m ∈ N and a unique idempotent e ∈ R such that a m − e ∈ P (R), where P (R) is the prime radical of R.MR(2010) Subject Classification: 16U99, 16E50.
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