Rings Whose Non-Invertible Elements Are Strongly Nil-Clean

We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of a commuting nilpotent and an idempotent. In addition, we examine those rings in which the non-units are uniquely nil-clean in the sense that they are a sum...

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Published inLobachevskii journal of mathematics Vol. 45; no. 10; pp. 4980 - 5001
Main Authors Danchev, P., Hasanzadeh, O., Javan, A., Moussavi, A.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.10.2024
Springer Nature B.V
Subjects
Algebra
Analysis
Geometry
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Sums
strongly nil-clean element (ring)
uniquely clean element (ring)
nil-clean element (ring)
uniquely nil-clean element (ring)
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ISSN1995-0802
1818-9962
DOI10.1134/S1995080224602649

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Summary:We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of a commuting nilpotent and an idempotent. In addition, we examine those rings in which the non-units are uniquely nil-clean in the sense that they are a sum of a nilpotent and an unique idempotent. In fact, we succeeded to prove that these two classes of rings can completely be characterized in terms of already well-studied and fully described sorts of rings.
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ISSN:1995-0802
1818-9962
DOI:10.1134/S1995080224602649