In the title mol-ecular salt, C(11)H(17)N(2) (+)·Cl(-), the piperazin-1-ium ring adopts a chair conformation with the aromatic ring in a pseudo-equatorial orientation. The dihedral angle between the benzene ring and the mean plane of the piperazin-1-ium ring is 51.22 (6)°. In the crystal, N-H⋯Cl hydrogen bonds link the mol-ecules into chains propagating in [100]. Weak C-H⋯π inter-actions also o...

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which...

Based on the work of Hahn, Baer, Ostrowski, Krull, Kaplansky and the Artin-Schreier theory, and stimulated by the paper [L] of S. Lang in 1953, the theory of real places and convex valuations has witnessed a remarkable development and has become a basic tool in the theory of ordered fields and real algebraic geometry. Surveys on this development can be found in [LAM] and [PC]. In this paper, we...

Classically, Gröbner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y]∗) and use this ...

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are cha...

Let $\mathcal R $ be a finite valuation ring of order $q^r$ with $q$ power an odd prime number, and let A \subset \mathcal $. We improve recent result due to Yazici (2018) on sum-product type problem. More precisely, we prove that