ئەز   Renas Tahsin M.Salim

A ring R is considered a strongly 2-nil clean ring, or (strongly 2-NC ring for short), if each element in R can be expressed as the sum of a nilpotent and two idempotents that commute with each other. In this paper, further properties of strongly 2-NC rings are given. Furthermore, we introduce and explore a special type of strongly 2-NC ring where every unit is of order 2, which we refer to as a strongly 2-NC rings with U(R) = 2. It was proved that the Jacobson radical over a strongly 2-NC ring is a nil ideal, here, we demonstrated that the Jacobson radical over strongly 2-NC ring with U(R) = 2 is a nil ideal of characteristic 4. We compare this ring with other rings, since every SNC ring is strongly 2-NC, but not every unit of order 2, and if R is a strongly 2-NC with U(R) = 2, then R need not be SNC ring. In order to get Nil(R) = 0, we added one more condition involving this ring.

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.

In this paper we introduce the notion of semi 𝜋𝜋 -regular clean rings. Some properties of semi 𝜋𝜋-regular clean ring are investigated, which generalize the well-known results of clean ring, and it’s connection with other rings are given