Throughout this paper, R is an associative ring; andN ,C,C(R), and J denote, respectively, the set of nilpotent elements, the center, the commutator ideal, and the Jacobson radical. An element x of R is called potent if xn = x for some positive integer n= n(x) > 1. A ring R is called periodic if for every x in R, xm = xn for some distinct positive integersm=m(x), n = n(x). A ring R is called we...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

To calculation non-bonded interaction of the [CoCl6]3- complex embedded in nano ring, we focus on the single wall boron-nitride B18N18 nano ring. Thus, the geometry of B18N18 nano ring has been optimized by B3LYP method with EPR-II (Electron paramagnetic resonance) basis set and geometry of the [CoCl6]3- complex has been optimized at B3LYP method with Aldrich’s VTZ basis set and Stuttgart RSC 1...