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Rings in which elements are the sum of a nilpotent and a root of a fixed polynomial that commute. (English) Zbl 1362.16025

Summary: An element in a ring \(R\) with identity is said to be strongly nil clean if it is the sum of an idempotent and a nilpotent that commute, \(R\) is said to be strongly nil clean if every element of \(R\) is strongly nil clean. Let \(C(R)\) be the center of a ring \(R\) and \(g(x)\) be a fixed polynomial in \(C(R)[x]\). Then \(R\) is said to be strongly \(g(x)\)-nil clean if every element in \(R\) is a sum of a nilpotent and a root of \(g(x)\) that commute. In this paper, we give some relations between strongly nil clean rings and strongly \(g(x)\)-nil clean rings. Various basic properties of strongly \(g(x)\)-nil cleans are proved and many examples are given.

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16U99 Conditions on elements
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