Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimes\Lambda[y_1,\cdots,y_n]$ by an ideal generated monomials with leading coefficients 1. Then we constructed space~$X_A$ such that is isomorphic to $H^*(X_A)$ modulo torsion elements.

We prove that in a ring Rwith an identity there exists an element a ∈ R and a nonzero derivation d ∈ Der R such that ad(a) 6= 0. A ring R is said to be a d-rigid ring for some derivation d ∈ Der R if d(a) = 0 or ad(a) 6= 0 for all a ∈ R. We study rings with rigid derivations and establish that a commutative Artinian ring R either has a non-rigid derivation or R = R1 ⊕ · · · ⊕ Rn is a ring direc...

The purpose of this short note is to prove that if R an alternative ring whose associators are not zero-divisors, then has no zero-divisors. By a result Bruck and Kleinfeld, if, in addition, the characteristic 2, central quotient octonion division algebra over some field.

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by action finite group $G$ and show that Balmer spectrum tensor triangulated category perfect complexes on homeomorphic to space homogeneous prime ideals in cohomology ring $H^*(G,A)$ .

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in Cn modulo a twisted action of the maximal torus in SL(n, C). We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst O(n). On the other hand, we show that the associated semigroup of Gelfand–Tsetlin patterns can have an essenti...

Let R be a commutative ring with identity and T (R) its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of co...

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

We study the ring Γ of all functions N → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the “truncations” Γn = { f ∈ Γ ∀m > n : f(m) = 0 } . Each Γn is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the o...