In the title compound, C(21)H(28)Cl(2)N(3)O(6), the nitronyl nitroxide ring displays a half-chair conformation, whereas the pyrrolidine ring has an envelope conformation. These two rings are twisted to each other with N-C-C-N torsion angles around the connecting C-C bond of 48.9 (6) and -127.0 (5)°. The benzene ring is nearly perpendicular to the pyrrolidine ring, with torsion angles around the...

In the crystal structure, the title Schiff base compound, C(23)H(30)N(2)O(4), exhibits crystallographic twofold rotation symmetry. The imino group is coplanar with the aromatic ring with an N-C-C-C torsion angle of -179.72 (9)°. An intra-molecular O-H⋯N hydrogen bond forms a six-membered ring, producing an S(6) ring motif. The dihedral angle between symmetry related benzene rings is 28.05 (5)°....

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [16], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This...

In this paper, we consider for any free presentation G = F/R of a group G the coinvariance H0(G,R ⊗n ab ) of the n-th tensor power of the relation module Rab and show that the homology group H2n(G,Z) may be identified with the limit of the groups H0(G,R ⊗n ab ), where the limit is taken over the category of these presentations of G. We also consider the free Lie ring generated by the relation m...

Let m,n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : R −→ R be a skew derivation of R and E(x) = D(xm+n+r)−D(xm)xn+r − xmD(xn)xr − xm+nD(xr). We prove that if E(x) = 0 for all x ∈ L, then D is a usual derivation of R or R satisfies s4(x1, . . . , x4), the standard identity of degree 4.

Let N be a prime Γ-near ring with multiplicative center Z. Let σ and τ be automorphisms of N and δ be a Γ− (σ, τ)-derivation of N such that N is 2-torsion free. In this paper the following results are proved: (1) If σγδ = δγσ and τγδ = δγτ and δ(N) ⊆ Z, or [δ(x), δ(y)]γ = 0, for all x, y ∈ N and γ ∈ Γ, then N is a commutative ring. (2) If δ1 is a Γ-derivation, δ2 is a Γ − (σ, τ) derivation of N...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

In the title compound, C(8)H(13)N(2)O(2) (+)·Cl(-)·H(2)O, the methyl C atom of the ethyl group is slightly out of the imidazole plane, with an N-C(ring)-C-C torsion angle of -15.1 (2)°. In the crystal structure, there are strong inter-molecular hydrogen-bonding inter-actions between the solvent water mol-ecule, the free chloride anion and the organic cation, resulting in a two-dimensional supra...

In the title mol-ecule, C(21)H(19)N(3)O(2), the central pyridine ring makes dihedral angles of 14.46 (9) and 34.67 (8)° with the 4-amino- and 4-eth-oxy-substituted benzene rings, respectively. The eth-oxy group is essentially coplanar with the attached benzene ring [C-O-C-C torsion angle = 178.70 (16)°] as is the meth-oxy group with the pyridine ring [C-O-C-N torsion angle = -3.0 (3)°]. In the ...

The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some...