In the title compound, C(14)H(14)F(4)N(2)O(3)S, the hexa-hydro-pyrimidine ring adopts a half-chair conformation. The mol-ecular conformation is stabilized by an intra-molecular O-H⋯O hydrogen bond, generating an S(6) ring. The crystal structure features O-H⋯S and N-H⋯S hydrogen bonds.

Theorem. Let R be a Cohen-Macaulay ring (locally, always) 1 c R an ideal o f height at least 2, S the Rees ring of R with respect to I, and G = S /S I the associated graded ring. Assume that S and G are Cohen-Macaulay rings, and that S has a canonical module cos. Then G has a canonical module r and: (i) I f co s can be embedded into S such that cos (considered as an ideal now) is not contained ...

Let S be a subring of the ring R. We investigate the question of whether S ∩ U(R) = U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.

Let (R; m) be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring S of R such that (i) S dominates R, (ii) the residue eld of S is a nite purely transcendental extension of R=m, (iii) dim(S) 1. In addition, it is shown that S can be obtained so that either mS is the maximal ideal of S or S is a localization of a nitely generated R-algebra.

In the title Schiff base compound, C(21)H(22)Cl(2)N(4)S, the triazole ring makes dihedral angles of 2.15 (11) and 87.48 (11)° with the 2,6-dichloro-phenyl and methyl-propyl-phenyl rings, respectively. Weak intra-molecular C-H⋯S and C-H⋯Cl inter-actions generate S(6) and S(5) ring motifs, respectively. In the crystal structure, centrosymmetrically related mol-ecules are linked into dimers by N-H...

In the title thione-Schiff base compound, C(10)H(9)N(3)O(2)S(2), the dihedral angle between the benzene ring and the five-membered ring is 6.69 (8)°. An intra-molecular O-H⋯N hydrogen bond forms an S(2) (2)(6) ring. In the crystal, inversion dimers linked by pairs of N-H⋯S inter-actions occur, generating R(2) (2)(8) ring motifs. The crystal structure features a S⋯S contact [3.3776 (7) Å], which...

A triangular matrix ring Λ is defined by a triplet (R, S, M) where R and S are rings and RMS is an S-R-bimodule. In the main theorem of this paper we show that if TS is a tilting S-module, then under certain homological conditions on the S-module MS , one can extend TS to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M ...

We next want to construct a much larger ring in which infinite sums of multiples of elements of S are allowed. In order to insure that multiplication is well-defined, from now on we assume that S has the following additional property: (#) For all s ∈ S, {(s1, s2) ∈ S × S : s1s2 = s} is finite. Thus, each element of S has only finitely many factorizations as a product of two elements. For exampl...

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, C14H14F4N2O3S, the central di-hydro-pyrimidine ring adopts a sofa conformation with the C atom bearing the 2-fluoro-benzene ring displaced by 0.596 (3) Å from the other five atoms. The 2-fluoro-benzene ring is positioned axially and bis-ects the pyrimidine ring with a dihedral angle of 70.92 (8)°. The mol-ecular conformation is stabilized by an intra-molecular O-H⋯O hydro...