Rings Whose Clean and Nil-Clean Elements Have Some Clean-Like Properties

2024/01/04 by Peter Danchev, Arash Javan, Danchev, Peter +3
Mathematics · #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2401.02189

Abstract

We define two types of rings, namely the so-called CSNC and NCUC that are those rings whose clean elements are strongly nil-clean, respectively, whose nil-clean elements are uniquely clean. Our results obtained in this paper somewhat expand these obtained by Calugareanu-Zhou in Mediterr. J. Math. (2023) and by Cui-Danchev-Jin in Publ. Math. Debrecen (2024), respectively.

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• Rings Whose Clean Elements Are Uniquely Strongly Clean
• Rings Whose Non-Invertible Elements are Weakly Nil-Clean
• Rings whose Nil-Clean and Clean Elements are Uniquely Nil-Clean
• Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean
• Rings whose Non-Invertible Elements are Strongly Nil-Clean