Let R be a discrete valuation ring with a unique maximal ideal πR and a quotient field K, and let F = R/πR be the residue class field. Let n ≥ 2 be an integer and {λij | 1 ≤ i, j ≤ n} a set of n integers satisfying λii = 0, λik + λkj ≥ λij , λij + λji > 0 (if i = j) for all 1 ≤ i, j, k ≤ n. Then Λ = (πijR) is a basic semiperfect Noetherian R-subalgebra of the full n× n matrix algebra Mn(K). We ...

Equatorial/axial conversion in piperidine and phosphorinane with different substituents wereinvestigated with great details. Three possible routes, i.e. heteroatom inversion and two ring inversiontype were considered. Ring conversion can occur via two pathways one starts with ring flatteningfrom the heteroatom site (nitrogen in piperidine and phosphorous in phosphorinane) and the otherinitiates...

‎let $r$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$u$ the utumi quotient ring of $r$‎, ‎$c$ $(=z(u))$ the extended centroid‎ ‎of $r$‎. ‎let $0neq ain r$ and $f(x_1,ldots,x_n)$ a multilinear‎ ‎polynomial over $c$ which is noncentral valued on $r$‎. ‎suppose‎ ‎that $g$ and $h$ are two nonzero generalized derivations of $r$‎ ‎such that $a(h(f(x))f(x)-f(x)g(f(x)))in ...

Let G be a finite group, and R be a commutative ring. This note proposes a generalization to any Green functor for G over R of the construction of the Hochschild cohomology ring HH∗(G,R) from the ordinary cohomology functor H∗(−, R). Another special case is the construction of the crossed Burnside ring of G from the ordinary Burnside functor. The general abstract setting is the following : let ...

The NMR spectra of azo dyes, 5-arylazobarbituric (5a-g), 5-arylazo-1,3-dimethylbarbituric(6a-g) and 5-arylazothiobarbituric acids (7a-g) were studied in DMSO-d6 in differentconcentrations. An intramolecular hydrogen bond was observed and indicating that thehydrazone forms is mostly predominant. The peak of the hydrazone proton was severelybroadened and its chemical shift appeared at down field ...

‎let $g$ be a group with identity $e.$ let $r$ be a $g$-graded‎ ‎commutative ring and $m$ a graded $r$-module‎. ‎in this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎for example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $m$ are studied‎.‎

w x In 8 , K. Morita introduced a useful notion of duality between categories of modules, usually called ‘‘Morita duality.’’ He proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules. On the other hand, the equivalences between categories of comodules over a coalgebra w x were characterized by M. Ta...

in this note we introduce the notion of weak mccoy rings as a generalization of mccoy rings, and investigate their properties. also we show that, if is a semi-commutative ring, then is weak mccoy if and only if is weak mccoy.

The origin of Saturn's narrow G ring has been unclear. We show that it contains a bright arc located 167,495.6 +/- 1.3 km from Saturn's center. This longitudinally localized material is trapped in a 7:6 corotation eccentricity resonance with the satellite Mimas. The cameras aboard the Cassini spacecraft mainly observe small (1 to 10 micrometers) dust grains in this region, but a sharp decrease ...