NE-Nil Clean Rings and Their Generalization

doi:10.13189/ms.2023.110417

Journals Information

Mathematics and Statistics Vol. 11(4), pp. 746 - 751
DOI: 10.13189/ms.2023.110417
Reprint (PDF) (251Kb)

NE-Nil Clean Rings and Their Generalization

Renas T. M.Salim ¹^,*, Nazar H. Shuker ²
¹ Department of Mathematics, Faculty of Science, Zakho University, Iraq
² Department of Math., College of Computer Science and Math., Mosul University, Iraq

ABSTRACT

This article presents the concept of a NE-nil clean ring, which is a generalization of the strongly nil clean ring. A ring R is considered NE-nil clean if, for every a in R, there exists a₁ in R such that aa₁ = with a − a₁ = q and a₁q = qa₁, where q is nilpotent and is idempotent. This article's aim is to introduce a new type of ring, the NE-nil clean ring, and provide the fundamental properties of this ring. We also establish the relationship between NE-nil clean rings and 2-Boolean rings. Additionally, we demonstrate that the Jacobson radical and the right singular ideal over NE-nil clean ring are nil ideals. Among other results, we prove that every strongly nil clean ring and every weak * nil clean ring are NE-nil clean. We establish that a strongly 2-nil clean ring and NE-nil clean ring are equivalent. Furthermore, we introduce and investigate NT-nil clean ring, that is a ring with every a in R, there exists a₁ in R such that aa₁ = t with a − a₁ = q and a₁q = qa₁, where t is a tripotent and q is nilpotent, by showing that these rings are a generalization of NE-nil clean rings. We provide the basic properties of these rings and explore their relationship with NE-nil clean and Zhou rings.

KEYWORDS
Idempotents, Tripotents, Strongly Nil Clean, Strongly 2-nil Clean

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Renas T. M.Salim , Nazar H. Shuker , "NE-Nil Clean Rings and Their Generalization," Mathematics and Statistics, Vol. 11, No. 4, pp. 746 - 751, 2023. DOI: 10.13189/ms.2023.110417.

(b). APA Format:
Renas T. M.Salim , Nazar H. Shuker (2023). NE-Nil Clean Rings and Their Generalization. Mathematics and Statistics, 11(4), 746 - 751. DOI: 10.13189/ms.2023.110417.