Generalized Cox’s construction associates with an algebraic variety a remarkable invariant – its total coordinate ring, or Cox ring. In this note we give a new proof of factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on a notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ri...

The crystal structure of the title compound, C(15)H(13)N(3)O, consists of columns of mol-ecules that are inter-connected by N-H⋯N hydrogen bonds in the direction of the b axis. The torsion angle between the imidazo[1,2-a]pyridine ring system and the phenyl ring is 9.04 (5)°.

In the title compound, C(9)H(11)NOS, the thia-zolidinyl ring is almost perpendicular to the phenyl ring with N-C-C-C torsion angles of 71.7 (2) and 107.1 (2)°. In the crystal, mol-ecules are connected via N-H⋯O and O-H⋯N hydrogen bonds, forming layers.

Let G be a split simple affine algebraic group of type A or C over a field k, and let E be a standard generic G-torsor over a field extension of k. We compute the Chow ring of the variety of Borel subgroups of G (also called the variety of complete flags of G), twisted by E. In most cases, the answer contains a large finite torsion subgroup. The torsion-free cases have been treated in the prede...

We study the free Lie ring of rank 2 in the variety of all centreby-nilpotent-by-abelian Lie rings of derived length 3. This is the quotient L/([γc(L′), L] + L′′′) with c > 2 where L is the free Lie ring of rank 2, γc(L′) is the c-th term of the lower central series of the derived ideal L′ of L, and L′′′ is the third term of the derived series of L. We show that the quotient γc(L′) + L′′′/[γc(L...

In the present article we discuss an approach to cohomological invariants of algebraic groups over fields characteristic zero based on Morava $K$-theories, which are generalized oriented cohomology theories in sense Levine--Morel. We show that second $K$-theory detects triviality Rost invariant and, more generally, relate and splitting motives. describe motives, compute some affine varieties,...

Let $I$ be a square-free monomial ideal in polynomial ring $R=K[x_1,\ldots, x_n]$ over field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ the graded maximal of $R$, and $\{u_1, u_{\beta_1(I)}\}$ independent set minimal generators such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(R/(I\setminus x_i)^t)$ for all $x_i\mid \prod_{i=1}^{\beta_1(I)}u_i$ some positive integer $t$, where $I\setminus x_...

Abstract: Let be a commutative ring and be a unitary module. We define a semiprime submodule of a module and consider various properties of it. Also we define semi-radical of a submodule of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.  

This research aims to give the decompositions of a finitely generated module over some special ring, such as principal ideal domain and Dedekind domain. One main problems with theory is analyze objects module. was using literature study on modules topics from scientific journals, especially those related theory. And by selective cases we find pattern build conjecture or hypothesis, deductive pr...