COMMUTATIVE RINGS WHOSE ELEMENTS ARE A SUM OF A UNIT AND IDEMPOTENT

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RINGS IN WHICH ELEMENTS ARE UNIQUELY THE SUM OF AN IDEMPOTENT AND A UNIT (2004, background): “(3)⇒(4) by Corollary 2; (4)⇒(1) is clear; and (2)⇒(3) because if a = e + u in R where e2 = e and u ∈ U, then a = 1 + e = 1 − e is an idempotent.”
Pseudo Drazin inverses in associative rings and Banach algebras (2012, background): “A ring is called uniquely clean if every element is uniquely the sum of an idempotent and a unit, which was discussed by Anderson–Camillo [1] and Nicholson–Zhou [21] in the commutative and noncommutative cases, respectively.”
Graphs Attached to Rings Revisited (2011, unknown): “[28, 1]).”
Commutative weakly nil clean unital rings (2015, background): “The notion of uniquely clean rings was firstly defined in [4] in the commutative case as those rings in which every element is uniquely the sum of a unit and an idempotent.”
On Strongly Clean Matrix and Triangular Matrix Rings (2006, background): “It was proven in Wang and Chen ( 2004 ) that if R is a commutative local ring for which every element is uniquely the sum of an idempotent and a unit (or equivalently, R / J ( R ) ≅ ℤ 2 by Anderson and Camillo ( 2002 , Corollary 22)), then 𝕋 n ( R ) is a strongly clean ring for every”