The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

If M is a simple module over a ring R then, by the Schur’s lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

The title compound, C(16)H(22)Cl(2), a derivative of β-himachalene, was semi-synthesized from natural essential oils of Cedrus atlantica. The mol-ecule is built up from two fused six- and seven-membered rings. The six-membered ring has a perfect chair conformation, whereas the seven-membered ring displays a screw boat conformation; the dihedral angle between the rings is 46.48 (9)°.

Let X be any partially ordered set, R any commutative ring, and T = I * (X,R) the weak incidence algebra of X over R. Let Z be a finite nonempty subset of X, L(Z) = {x ∈ X : x z for some z ∈ Z}, and M = Te Z. Various chain conditions on M are investigated. The results so proved are used to construct some classes of right perfect rings that are not left perfect.

Density functional theory calculations were performed to investigate the endohedral and exohedral adsorption of a H2 molecule on the classical and nonclassical fullerenes Cx (x=58, 59, 60, 62) with seven-, eight-, and ninemembered rings. The amplitude of adsorption energies are within 0.03eV and the molecule-fullerene interaction belongs to van der Waals type. Penetration of a H2 molecule throu...

We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in categorical tilting context beyond. In n-tilting–cotilting correspondence situation, if $$\mathsf A$$ is a Grothendieck abelian category related B$$ equivalent contramodules over topological ring $$\mathfrak R$$ belonging one certain four classes rings (e. g., commutative), then l...

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...

Let A be a nondegenerate dimer (or ghor) algebra on torus, and let Z its center. Using cyclic contractions, we show the following are equivalent: is noetherian; noncommutative crepant resolution; each arrow of contained in perfect matching whose complement supports simple module; vertex corner rings eiAei pairwise isomorphic.

we introduce the notion ofstrongly $alpha$-reversible rings which is a strong version of$alpha$-reversible rings, and investigate its properties. we firstgive an example to show that strongly reversible rings need not bestrongly $alpha$-reversible. we next argue about the strong$alpha$-reversibility of some kinds of extensions. a number ofproperties of this version are established. it is shown ...

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete fields K 1 and 2 with perfect residue fields, show that if n -th rings are isomorphic for each ? , then isometric isomorphic. More generally, there is depending only on ramification indices such homomorphism from ring to can be lifted a between valuation rings. Moreover...