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Rings in which the power of every element is the sum of an idempotent and a unit. (English) Zbl 1499.16057

Summary: A ring \(R\) is uniquely \(\pi\)-clean if the power of every element can be uniquely written as the sum of an idempotent and a unit. We prove that a ring \(R\) is uniquely \(\pi\)-clean if and only if for any \(a\in R\), there exists an integer \(m\) and a central idempotent \(e\in R\) such that \(a^m -e\in J(R)\), if and only if \(R\) is abelian; idempotents lift modulo \(J(R)\); and \(R/P\) is torsion for all prime ideals \(P\supseteq J(R)\). Finally, we completely determine when a uniquely \(\pi\)-clean ring has nil Jacobson radical.

MSC:

16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
16U99 Conditions on elements
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
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References:

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