Let M be a 2-torsion free prime Γ-ring satisfying the condition a α b β c=a β b α c,∀a,b,c∈M and α,β∈Γ, U be an admissible Lie ideal of M and F=(f i ) i∈N be a generalized higher (U,M)-derivation of M with an associated higher (U,M)-derivation D=(d i ) i∈N of M. Then for all n∈N we prove that [Formula: see text]. Mathematics Subject Classification (2010): 13N15; 16W10; 17C50.

The title mol-ecule, C(10)H(8)ClN(3)O(7), is twisted with the dihedral angle between the amide and benzene ring being 38.75 (11)°. The C-N-C-C torsion angle between the amide and acetyl groups is -150.1 (2)°. Finally, each nitro group is twisted out of the plane of the benzene ring to which it is connected [O-N-C-C torsion angles = 34.0 (3) and -64.5 (3)°]. Linear supra-molecular chains along [...

In the title compound, C(8)H(7)ClN(4)S(2), the dihedral angle between the thia-zolidine ring (r.m.s. deviation = 0.028 Å) and the thia-zole ring (r.m.s. deviation = 0.004 Å) is 74.74 (6)°. The formonitrile group is almost coplanar with the attached ring [C-N-C-N torsion angle = 167 (2)°.

In the title compound, C(8)H(3)F(4)NO(3), the carb-oxy group lies nearly in the plane of the ring with a C-C-C-O torsion angle of -10.5 (4)°. The carbamoyl group is almost perpendic-ular to the benzene ring [C-C-C-O torsion angle = 82.2 (4) °]. In the crystal, molecules are linked via O-H⋯O and N-H⋯O hydrogen bonds involving the carbamoyl and carb-oxy groups.

The title compound, C(19)H(16)N(4)O(2), has an almost planar fused N-heterocyclic system (r.m.s. deviation = 0.031 Å) and an almost planar five-membered 1,3-oxazolidine ring (r.m.s. deviation = 0.015 Å) at the ends of an ethyl-ene chain [N-C-C-N torsion angle = -65.6 (2)°]. The ring systems are inclined at 38.1 (1)° to one another.

We identify some classes $$\mathcal {C}$$ of mixed groups such that if $$G\in \mathcal has the cancellation property then Walk-endomorphism ring G unit lifting property. In particular, is a self-small group torsion-free rank at most 4 with it decomposition $$G=F\oplus H$$ F free and H

Abstract Fix an odd prime $p$. The results in this paper are modeled after work of Hesselholt and Hesselholt–Madsen on the $p$-typical absolute de Rham–Witt complex mixed characteristic. We have two primary results. 1st result is exact sequence that describes kernel restriction map over $A$, where $A$ ring integers algebraic extension $\textbf{Q}_p$ or a $p$-torsion-free perfectoid ring. 2nd de...

By a theorem of Elman and Lam, fields over which quadratic forms are classified by the classical invariants dimension, signed discriminant, Clifford invariant and signatures are exactly those fields F for which the third power IF of the fundamental ideal IF in the Witt ring WF is torsion free. We study the possible values of the uinvariant (resp. the Hasse number ũ) of such fields, i.e. the sup...