Let M be a 2 and 3-torsion free prime Γ-ring, d a nonzero derivation on M and U a nonzero Lie ideal of M . In this paper it is proved that U is a central Lie ideal of M if d satisfies one of the following (i) d(U) ⊂ Z, (ii) d(U) ⊂ U and d(U) = 0, (iii) d(U) ⊂ U , d(U) ⊂ Z.

We prove that a Lie nilpotent one-sided ideal of an associative ring R is contained in a Lie solvable two-sided ideal of R. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of R. One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form [. . . [[r1, r2], . . .], rn−1...

We study the free Lie ring of rank 2 in the variety of all centreby-nilpotent-by-abelian Lie rings of derived length 3. This is the quotient L/([γc(L′), L] + L′′′) with c > 2 where L is the free Lie ring of rank 2, γc(L′) is the c-th term of the lower central series of the derived ideal L′ of L, and L′′′ is the third term of the derived series of L. We show that the quotient γc(L′) + L′′′/[γc(L...

The title compound, C(13)H(13)N(3), is non-planar, with the pyridine and phenyl rings inclined at an angle of 80.7 (3)°. The central ethyl-idenehydrazine atoms lie in a plane [mean deviation = 0.013 (1) Å], which forms dihedral angles of 88.5 (1) and 9.4 (1)° with the pyridine and phenyl rings, respectively. In the crystal structure, mol-ecules are linked by inter-molecular N-H⋯N hydrogen bonds...

In the title compound, C15H15N3OS, the dihedral angle between the planes of the benzene and pyridine rings is 26.86 (9)°. Intra-molecular N-H⋯O and C-H⋯S hydrogen bonds both generate S(6) rings. The C=O and C=S bonds lie to opposite sides of the mol-ecule. In the crystal, inversion dimers linked by pairs of N-H⋯S hydrogen bonds generate R 2 (2)(8) loops.

In the title compound, C(23)H(27)NO(4), the piperidine ring adopts a distorted boat conformation. The meth-oxy groups lie in the plane of benzene rings to which they are attached [maximum deviations of 0.014 (3) and 0.007 (3) Å]. The benzene rings are oriented at angles of 67.2 (1) and 87.0 (1)° with respect to the best plane through the four co-planar atoms of the piperidine ring.

The two independent mol-ecules of the title compound, C(7)H(5)ClN(4), both lie on a twofold rotation axis that passes through the centroids of the five- and six-membered rings and the attached Cl C atom. One molecule is nearly planar [dihedral angle between rings = 0.22 (6)°], whereas the other is significantly twisted [dihedral angle = 17.38 (6)°]. In the crystal, adjacent mol-ecules are linke...

In the title compound, C(22)H(23)Cl(2)NO(4), the piperidine ring adopts a distorted boat conformation. The meth-oxy groups lie in the plane of the benzene rings to which they are attached. The benzene rings are oriented at angles of 84.3 (1) and 76.8 (1)° with respect to the best plane through the piperidine ring. The crystal packing is stabilized by intermolecular C-H⋯O inter-actions.

In the title compound, C(26)H(20)N(2)O(4), the central isoindole core is almost planar (r.m.s. deviation = 0.043 Å). The phenyl rings lie to either side of the plane [dihedral angles = 88.64 (5) and 67.74 (6)°] and the dihedral angle between the phenyl rings is 63.39 (7)°. In the crystal, mol-ecules are linked by C-H⋯O inter-actions; notably, one carbonyl O atom accepts three such bonds.