let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

In the title compound, C(20)H(17)NO(4), the double bond of the acrylonitrile group separating the 1-benzofuran moiety from the 3,4,5-trimeth-oxy-phenyl ring has Z geometry. The 1-benzofuran groups are π-π stacked with inversion-related counterparts such that the furan ring centroid-centroid distance is 3.804 (5) Å. The dihedral angle between the planes of the trimeth-oxy-phenyl ring and the acr...

The title azo mol-ecule, C16H9Br3N2O, adopts a trans conformation with respect to the azo N=N double bond. An intra-molecular O-H⋯N hydrogen bond forms an S(6) ring motif. The dihedral angle between the naphthalene ring system and the benzene ring is 33.80 (16)°. In the crystal, mol-ecules are stacked in columns along the a axis by π-π inter-actions [centroid-centroid distances = 3.815 (3) and ...

In the crystal structure of the title compound, C(20)H(16)O(5), the mol-ecule assumes an E configuration with the benzene ring and chromenecarboxyl group located on opposite ends of the C=C double bond. The chromene ring system and benzene ring are oriented at a dihedral angle of 74.66 (12)°. Weak inter-molecular C-H⋯O hydrogen bonding is present in the crystal structure.