In the title compound, C9H9ClN4OS, the dihedral angle between the acetamide moiety and the pyridine ring is 4.83 (12)°. The O=C-C-Cl torsion angle is 46.4 (3)° and an intra-molecular C-H⋯O inter-action generates an S(6) ring. In the crystal, mol-ecules are linked by N-H⋯O, N-H⋯N and C-H⋯N hydrogen bonds, generating sheets lying parallel to (120).

The pyrrolo-pyridine system in the title compound, C(27)H(29)N(5)O(4)S, is oriented at a dihedral angle of 71.20 (5)° towards the phenyl ring of the tosyl residue and at a dihedral angle of 45.43 (4)° towards the benzyl group. The structure shows an intra-molecular N-H⋯O and a weak intra-molecular N-H⋯N hydrogen bond. The piperidine ring adopts a chair conformation, with the cis substituents di...

In the title compound, C(18)H(15)ClFN(3)O(4), the dihedral angle between the substituted pyridine ring and the oxadiazo-line ring is 9.73 (19)° and the acyl group is coplanar with the oxadiazo-line ring [O-C-N-C torsion angle = -2.1 (3)°]. Furthermore, the substituted benzene ring is almost orthogonal with the oxadiazo-line ring, the dihedral angle between them being 87.56 (18)°.

Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to Kn for some n > 0. When we replace the scalar field K with a commutative ring A, it is no longer true that every A-module with a finite generating set has a basis: not all modules have bases. But when A is a PID, we get something nearly as good as that: (1) Every submodule of An has a basis ...

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ torsion-free group, then the only units of group ring $K[G]$ are trivial units, is, non-zero scalar multiples elements. We give concrete counterexample this conjecture; virtually abelian order two.

we call a module  essentially retractable if homr for all essential submodules n of m. for a right fbn ring r, it is shown that: (i)  a non-zero module  is retractable (in the sense that homr for all non-zero ) if and only if certain factor modules of m are essentially retractable nonsingular modules over r modulo their annihilators. (ii)  a non-zero module  is essentially retractable if and on...

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a bas...

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorp...

Several of the fundamental theorems about algebraic K, and Kr are concerned with finding unimodular elements, that is, elements of a projective module which generate a free summand. In this paper we use the notion of a basic element (in the terminology of Swan [22]) to extend these theorems to the context of finitely generated modules. Our techniques allow a simplification and strengthening of ...

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a bas...