Let $R$ be a regular local ring. $\mathbf {G}$ reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that principal ${\mathbf {G}}$-bundle over is trivial if it the quotient field $R$. The known when contains field. We prove for large class rings

In 1999, Reg Wood conjectured that the quotient of Q[x1, . . . , xn] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group Sn. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the ...

In 1982 Richard P. Stanley conjectured that any finitely generated Zn-graded module M over a finitely generated Nn-graded K-algebra R can be decomposed as a direct sum M = ⊕t i=1 νi Si of finitely many free modules νi Si which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least...

We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator ideals $\mathcal I$ contained in ideal of compact operators, where $X$ is a space that fails I$-approximation property. The main results concern nilpotent algebras $\mathfrak A_X^{\mathcal{QN}_p}$ and A_X^{\mathcal{SK}_p}$ quasi $p$-nuclear opera...

In this paper, persents the definitions of strongly prime ideal, strongly prime N-subgroup, Pseudo-valuation near ring and Pseudo-valuation N-group. Some of their properties have also been proven by theorems. Then it is shown that, if N be near ring with quotient near-field K and P be a strongly prime ideal of near ring N, then is a strongly prime ideal of ‎‎, for any multiplication subset S of...

Abstract We study the structure of constacyclic codes of length ps over the ring Fpk + uFpk + · · · + um−1Fpk in terms of ideals in the quotient ring (Fpk + uFpk + · · · + um−1Fpk)[x]/〈xp s − λ〉, where λ is a unit element in Fpk + uFpk + · · · + um−1Fpk . In particular, we complete a structural classification of constacyclic codes of length ps over Fpk + uFpk and determine their minimum Hamming...

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere g-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s manifold g-c...

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere gconjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s manifold g-co...

We obtain a new important basic result on splice-quotient singularities in an elegant combinatorial-geometric way: every level of the divisorial filtration of the ring of functions is generated by monomials of the defining coordinate functions. The elegant way is the language of of line bundles based on Okuma’s description of the function ring of the universal abelian cover. As an easy applicat...

The algebraic structure of the set of doubly stochastic circulants is that of a semi-ring. The concept of a prime in the doubly stochastic circulants is introduced in this paper and examples are given. The classiication of a prime in the doubly stochastic circulants is equivalent to the solvability of a linear equation over a doubly stochastic circulant. A representation of doubly stochastic ci...