We introduce the concept of a weak idempotent Nil-clean ring as a generalization of a weakly Nil-clean ring. We give certain characterizations for weak idempotent Nil-clean rings in terms of Jacobson radical and Nil radical. Further, we obtain any weak idempotent Nil-clean ring is a direct product Nil clean rings in terms of Jacobson radicals .

‎We introduce the concept of weakly Nil‎ - ‎semicommutative or WNSC rings and provide a condition that establishes the equivalence of WNSC rings to three generalised classes of semicommutative rings‎. ‎We prove the equivalence between WNSC Laurent polynomial rings and WNSC polynomial rings‎. ‎We supply examples of these classes of rings by considering Nagata and Dorroh extensions‎. ‎We also give a characterization for a ring of Morita context with zero pairings to be WNSC‎.

We introduce the concept of a weak idempotent Nil-clean ring as a generalization of a weakly Nil-clean ring. We give certain characterizations for weak idempotent Nil-clean rings in terms of Jacobson radical and Nil radical. Further, we obtain any weak idempotent Nil-clean ring is a direct product Nil clean rings in terms of Jacobson radicals .

Water and sediment samples were collected from six different stations, located along the Nil River between February and June 2015. Concentrations of cadmium, lead, zinc, and copper were determined. The extent of the sediment pollution was assessed, using the multiple pollution indices, namely contamination factor (CF), pollution load index (PLI), and the geoaccumulation index (Igeo). The results showed that the level of metals in water samples exceeded background concentrations for Cd and Pb, and the average values for those elements were higher than those of Zn and Cu, ranked as the following: Pb (0. 58)> Zn (0. 38)> Cd (0. 32)> Cu (0. 061). For sedimentary samples, the results showed that Zn and Pb concentrations were greater than the concentrations of Cu and Cd, exceeding the background values (except for Cu). The concentration of the tested heavy metals decreased to the following order: Zn (96. 2)> Pb (61. 5)> Cu (38. 83)> Cd (2. 34). The Igeo values revealed that Cd (2. 87) and Pb (1. 61) had accumulated significantly in the Nil River. Contamination factor (CF) confirmed that the sedimentary samples were moderate to very high in terms of Cd, Pb, and Zn contamination. The Pollution Load Index (PLI) values were above one (>1), indicating an advanced decline of the sediment quality. Also, the results showed that the stations, located at the middle portion of the River (S3, S4, and S5) have higher levels of metals than the other stations (S1, S2, and S6). The assessment of heavy metals' levels in water and sedimentary samples indicated that River water and the sediments in the study area were strongly impacted by agricultural activities and domestic waste water.

We introduce the concept of strongly weak idempotent Nil-clean rings which is a generalization of strongly weakly Nil clean rings. We characterize strongly weak idempotent Nil-clean rings in terms of the set of Nilpotent elements, homomorphic images, and Jacobson radicals. We prove that a ring $R$ is strongly weak idempotent Nil-clean if and only if for any $a\in R$, $a-a^3$ is Nilpotent if and only if $Nil(R)$ forms an ideal and $R/{Nil(R)}$ is reduced weak idempotent Nil-clean if and only if $R$ has no homomorphic image $\mathbb{Z}_3\oplus \mathbb{Z}_3$ and $a^2-a^4$ is Nilpotent. Moreover, we prove that a strongly weak idempotent Nil-clean ring $R$ with $2\in J(R)$ satisfies Nil-involution property.

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In this paper, we study the structure of the set of Nilpotent elements in Armendariz rings of Hurwitz series type and introduce Nil Armendariz as a generalization. It is proved that a ring $R$ is Nil Armendariz of Hurwitz series type if and only if $R$ has characteristic zero and $Nil(R)$ is an ideal. We provide many examples of Nil Armendariz rings of Hurwitz series type and extend the class of Nil Armendariz rings of Hurwitz series type through various ring extensions.