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...

Mixed fault diameter of a graph G, D(a,b)(G), is the maximal diameter of G after deletion of any a vertices and any b edges. Special cases are the (vertex) fault diameter D a = D(a,0) and the edge fault diameter D E a = D(0,a). Let G be a Cartesian graph bundle with fibre F over the base graph B. We show that (1) D a+b+1(G) ≤ D V a (F )+D V b (B) when the graphs F and B are kF -connected and kB...

KF/Al2O3 as a green and efficient catalyst has been used for the synthesis of mono and bis pyrimidine derivatives via three-component reaction of amidines, malononitrile, and aldehydes in EtOH at reflux. The great advantage of this catalyst is the ease of handling. KF/Al2O3 can be used and removed by filtration, avoid cu...

Let N ≡ 1(mod 4) be a positive integer and let be the single even primitive quadratic Dirichlet character on (Z/NZ)×. Let f ∈ S2(Γ0(N), ) be a newform with nebentypus . By the Shimura construction, f corresponds to an abelian variety Af defined over Q whose dimension is [Kf : Q] where Kf is the number field associated with f . When dimAf = 2, the Fricke involution wN acts on Af and is defined o...