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 = δ with a − a1 = q
and a1q = qa1, where q is nilpotent and δ 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 J (R) and the right singular ideal γ(R)
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.
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