We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and triangular $T_n(F_2)$ over two elements field $F_2$ are either n-torsion clean or almost clean, respectively. These results somewhat address settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well they supply more precise aspect nil-cleanness property of $n\times n$ all ...

Let R be a commutative ring with multiplicative identity and C coassociative counital R-coalgebra the α-condition. A clean comodules defined based on cleanness rings modules. C-comodule M is comodule if endomorphism of clean. over itself i.e., as For an idempotent e ∈ R, there are relations between eRe R. It’s motivated us to investigate this condition for coalgebra. any C, we can construct ⇀C↼...

The lowest eigenenergies of few, strongly interacting electrons in a one–dimensional ring are studied in presence of an impurity barrier. The persistent current I , periodic in an Aharonov–Bohm flux penetrating the ring, is strongly influenced by the electron spin. The impurity does not remove discontinuities in I at zero temperature. The total electron spin of the ground state oscillates with ...

Air exhibiting very low concentrations of light-scattering aerosol particles is occasionally observed even in industrialized regions. Evidence is presented that this very clean air results from highly efficient removal of aerosol particles coupled to the removal of water in precipitation.