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(32)H(28)N(2)O(2), the pyrrolidine ring adopts an envelope conformation, whereas the cyclo-hexa-none ring in the tetra-hydro-naphthalene fused-ring system adopts a half-chair conformation. The oxindole ring system is oriented at an angle of 48.2 (1)° with respect to the naphthyl ring system. An intra-molecular C-H⋯O close contact is observed. In the crystal, mol-ecules a...

The title compound, C(20)H(15)N(3)OS, is a 1H-pyrazol-3(2H)-one having aromatic 4-(1,3-benzothia-zol-2-yl) and 2-phenyl substit-uents. The five-membered ring and the fused-ring system are close to planar, the r.m.s. deviations being 0.025 and 0.005 Å, respectively. The five-membered ring is aligned at 67.5 (1)° with respect to the phenyl ring and at 4.7 (1)° with respect to the fused-ring syste...

In the title compound, C(38)H(32)N(2)O(2), the pyrrolidine ring adopts an envelope conformation, whereas the cyclo-hexa-none ring in the tetra-hydro-naphthalene fused-ring system adopts a half-chair conformation. The benzyl ring is oriented at an angle of 67.1 (1)° with respect to the naphthyl ring system. Four intra-molecular C-H⋯O close contacts and C-H⋯π inter-action are observed. In the cry...

In the title compound, C37H34N2O7, both pyrrolidine rings adopt envelope conformations. The β-lactam ring is close to planar (r.m.s. deviation = 0.0395 Å) and makes a dihedral angle of 83.35 (15)° with the furan ring. The O atom attached to the β-lactam ring deviates by 0.187 (2) Å from the mean plane of the ring. The β-lactam ring makes dihedral angles of 14.90 (15) and 27.72 (17)° with the me...

in this work, we investigate the transfer of some homological properties from a ring $r$ to its amalgamated duplication along some ideal $i$ of $r$ $rbowtie i$, and then generate new and original families of rings with these properties.

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...

Cytokinesis involves constriction of a contractile actomyosin ring. The mechanisms generating ring tension and setting the constriction rate remain unknown because the organization of the ring is poorly characterized, its tension was rarely measured, and constriction is coupled to other processes. To isolate ring mechanisms, we studied fission yeast protoplasts, in which constriction occurs wit...