In the title compound, C(27)H(30)O(8), the cyclo-hexane ring is in a chair conformation, while the five-membered ring adopts an envelope conformation. The 1,3-dioxane ring is oriented with respect to the benzene rings at dihedral angles of 53.38 (3) and 55.31 (3)°, while the dihedral angle between the benzene rings is 71.56 (3)°. In the crystal structure, inter-molecular C-H⋯O inter-actions lin...

In the title compound, C(20)H(27)N(3)O(4), both cyclo-hexane rings adopt chair conformations. The benzene ring and the amide group are oriented at a dihedral angle of 62.1 (2)°. In the crystal structure, inter-molecular N-H⋯O and C-H⋯O hydrogen bonds link the mol-ecules into chains propagating in [010], which contain R(2) (2)(12) ring motifs.

In the mol-ecule of the title compound, C(20)H(25)N(3)O(3), the aromatic rings are oriented at a dihedral angle of 88.36 (3)°. The cyclo-hexane ring adopts a chair conformation. In the crystal structure, inter-molecular N-H⋯O and O-H⋯N hydrogen bonds link the mol-ecules. C-H⋯π inter-actions are also present.

Let $R$ be a prime ring with extended centroid $C$, $H$ a generalized derivation of $R$ and $ngeq 1$ a fixed integer. In this paper we study the situations: (1) If $(H(xy))^n =(H(x))^n(H(y))^n$ for all $x,yin R$; (2) obtain some related result in case $R$ is a noncommutative Banach algebra and $H$ is continuous or spectrally bounded.

Replacement of phenyl ring(s) in tetraphenylethene by naphthalene ring(s) generates a series of new luminogens with aggregation-induced emission (AIE) characteristics, demonstrating that bulky naphthalene rings can serve as a rotor to construct AIE luminogens.

In the deca-hydro-phenanthrenone ring system of the title compound, C(27)H(44)O, the two cyclo-hexane rings adopt chair conformations, whereas the cyclo-hexene ring adopts an envelope conformation. The cyclo-pentane ring is twisted. In the crystal structure, mol-ecules are stacked along the a axis, but no significant inter-molecular inter-actions are observed.

In the title compound, C22H27BrO3, the cyclo-hexane ring adopts a chair conformation. The dihedral angle between the benzene rings is 41.9 (4)°. In the crystal, mol-ecules are linked by O-H⋯O and C-H⋯O hydrogen bonds, forming a three-dimensional network. In addition, π-π stacking inter-actions [centroid-centroid distance = 3.953 (6) Å] between the benzene rings of the meth-oxy-benzene groups oc...

In the title racemic compound, C(14)H(10)ClNO(3), which contains four stereogenic centres, the cyclo-hexane ring tends towards a boat conformation, while the tetra-hydro-furan and dihydro-furan rings adopt envelope conformations. The dihedral angle between the mean planes of the pyrrolidine-2,5-dione unit and the 4-chloro-phenyl ring is 49.0 (2)°.

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

the concept of fuzzy soft γ-ring is introduced; and some properties of fuzzy soft γ-rings are given. then the definitions of fuzzy soft γ-ideals are proposed and some of their theories are considered.