As usual, Z, Q and C stand for the ring of integers, the field of rational numbers and the field of complex numbers respectively. If l is a prime then we write Fl,Zl and Ql for the l-element (finite) field, the ring of l-adic integers and field of l-adic numbers respectively. If A is a finite set then we write #(A) for the number of its elements. Let K be a field of characteristic different fro...

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F (x) of degree n in A[x] such that A ⊗k k[x](x)/(F (x)) is a free A–module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x). Intr...

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J . Let F = {In}n≥0 be a good filtration of ideals in A. Denote by FJ (F) = ⊕ n≥0 (In/JIn)t n the fiber cone of F with respect to J. The paper characterizes the multiplicity and the CohenMacaulayness of FJ (F) in terms of minimal reductions of F .

Let (A,m) be a Noetherian local ring and F = (In)n≥0 a filtration. In this paper, we study the Gorenstein properties of the fiber cone F (F), where F is a Hilbert filtration. Suppose that F (F) and G(F) are CohenMacaulay. If in addition, the associated graded ring G(F) is Gorenstein; similarly to the I-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal nu...

In the title compound, C26H22N2O, one phenyl ring, one five-membered N-heterocyclic ring and one six-membered carbocyclic ring make up the hexa-hydro-benzo[f]iso-indole core. Another phenyl group is attached to the heterocyclic N atom as a substituent. The non-aromatic five- and six-membered rings both exhibit boat conformations. In the crystal, weak C-H⋯O and C-H⋯N inter-actions establish the ...

In the title compound, C15H15FN2OS, the dihedral angle between the planes of the benzo-thio-phene ring system and the fluoro-benzene ring is 3.74 (14)°. The six-membered ring of the benzo-thio-phene moiety adopts a half-chair conformation. The mol-ecular conformation is consolidated by intra-molecular N-H⋯F and N-H⋯O hydrogen bonds. In the crystal, mol-ecules are linked by N-H⋯O hydrogen bonds,...

In 1971, Ruzsa conjectured that if f : N ? Z with ( n + k ) ? mod for every , ? and = O ? < e then is a polynomial. this paper, we investigate the analogous problem ring of polynomials over finite field using Polynomial Method in combinatorics.

In the title compound, C(20)H(19)FN(6)S, the pyrazole ring has an envelope conformation, with the methine C atom being the flap atom. The dihedral angle between the least-squares plane through the pyrazole and triazole rings is 7.59 (9)°, and the triazole and attached benzene ring form a dihedral angle of 74.79 (9)°. The thio-urea group is coplanar with the pyrazole ring [N-N-C-S torsion angle ...

Let u1×n, Xn×n, and vn×1 be matrices of indeterminates, AdjX be the classical adjoint of X, and H(n) be the ideal I1(uX)+ I1(Xv)+ I1(vu−AdjX). Vasconcelos has conjectured that H(n) is a perfect Gorenstein ideal of grade 2n. In this paper, we obtain the minimal free resolution of H(n); and thereby establish Vasconcelos’ conjecture. Let u1×n, Xn×n, and vn×1 be matrices of indeterminates over a co...

In the title compound, C(20)H(11)N(5)·CH(3)OH, the benzene ring is twisted by a small dihedral angle of 1.89 (11)° with respect to the imidazo[4,5-f][1,10]phenanthroline ring system. N-H⋯O and O-H⋯N hydrogen bonding is present in the crystal structure.