Clean general rings

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Commutative weakly nil clean unital rings (2015, background): “In a subsequent paper, [25], the same authors coined the term semi-boolean ring as a (general) ring I satisfying the condition that for all r ∈ I there is an e ∈ Id(R) and j ∈ J(R) such that r = j + e; this was done in the context of rings not necessarily commutative nor possessing an identity.”