let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuat...

Let V be an n-dimensional left vector space over a division ring R and n ≥ 3. Denote by Gk the Grassmann space of k-dimensional subspaces of V and put Gk for the set of all pairs (S, U) ∈ Gk ×Gn−k such that S+U = V . We study bijective transformations of Gk preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of V to itself or to the dual spac...

A ring R is uniquely (nil) clean in case for any $a in R$ there exists a uniquely idempotent $ein R$ such that $a-e$ is invertible (nilpotent). Let $C =(A V W B)$ be the Morita Context ring. We determine conditions under which the rings $A,B$ are uniquely (nil) clean. Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.

Abstract. Let R be a 2-torsion free ring with identity. In this paper, first we prove that any Jordan left derivation (hence, any left derivation) on the full matrix ringMn(R) (n 2) is identically zero, and any generalized left derivation on this ring is a right centralizer. Next, we show that if R is also a prime ring and n 1, then any Jordan left derivation on the ring Tn(R) of all n×n uppe...

The title compound, C15H12N2O4S, is V-shaped with the isoindoline ring system (r.m.s. deviation = 0.006 Å) inclined to the benzene ring by 84.27 (13)°. In the crystal, inversion dimers are formed via pairwise N-H⋯O hydrogen bonds. These dimers associate further into corrugated ribbons, via pairwise N-H⋯O and C-H⋯O hydrogen bonds, propagating along the a-axis direction and lying parallel to (001).

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) generated by {xixj;{i,j}∈E}. The conjecture we treat following.Conjecture 1. If has 2-linear resolution, then projective dimension of K[G], pd (k[G]), equals maximal degree vertex in G.As far as know, this first mentioned paper Gitler Valencia [7, Conjecture 4.13], there it called Eliahou-Villarreal conjecture. trea...

In the title solvate, C(11)H(11)N(3)O(2)S·CH(3)NO(2), the dihedral angle between the benzene ring and the N-containing ring is 85.94 (11)°, and an approximate V shape arises for the sulfonamide mol-ecule. In the crystal, N-H⋯O and N-H⋯N hydrogen bonds and weak C-H⋯O inter-actions link the sulfonamide mol-ecules into a three-dimensional network. The nitro-methane solvent mol-ecules are located i...

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...