‎Suppose $n$ is a fixed positive integer‎. ‎We introduce the relative n-th non-commuting graph $Gamma^{n} _{H,G}$‎, ‎associated to the non-abelian subgroup $H$ of group $G$‎. ‎The vertex set is $Gsetminus C^n_{H,G}$ in which $C^n_{H,G} = {xin G‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin H}$‎. ‎Moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $H$ and $xy^{n}eq y^{n}x$ or $x...

Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-...

It is shown that every  almost linear bijection $h : Arightarrow B$ of a unital $C^*$-algebra $A$ onto a unital$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in A$, all $y in A$, and all $nin mathbb Z$, andthat almost linear continuous bijection $h : A rightarrow B$ of aunital $C^*$-algebra $A$ of real rank zero onto a unital$C^*$-algebra...

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In the title compound, C(11)H(11)NO(2), the mean planes formed by the benzene ring and the C and N atoms of the acryl group are almost orthogonal to each other, with a dihedral angle of 85.7 (1)°. During the structure analysis, it was observed that the unit cell contains large accessible voids, with a volume of 186.9 Å(3), which may host disordered solvent mol-ecules. This affects the diffracti...

In the title complex, [CuCl(2)(C(24)H(24)N(2)O(2))], the Cu(II) cation is N,N',O-chelated by a 2,2'-(1,1'-binaphthyl-2,2'-diyldiimino)-diethanol ligand and coordinated by two chloride anions in a distorted square-pyramidal geometry. In the diethanol ligand, the two naphthalene ring systems are twisted with respect to each other at a dihedral angle of 68.30 (9)°. The uncoord-inated hy-droxy grou...