Let $mathcal{A}$ be a $C^*$-algebra and $Z(mathcal{A})$ the‎ ‎center of $mathcal{A}$‎. ‎A sequence ${L_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{A}$ with $L_{0}=I$‎, ‎where $I$ is the‎ ‎identity mapping‎ ‎on $mathcal{A}$‎, ‎is called a Lie higher derivation if‎ ‎$L_{n}[x,y]=sum_{i+j=n} [L_{i}x,L_{j}y]$ for all $x,y in  ‎mathcal{A}$ and all $ngeqslant0$‎. ‎We show that‎ ‎${L_{n}}_{n...

In this article, a log del Pezzo surface of index two means a projective normal non-Gorenstein surface S such that (S, 0) is a log-terminal pair, the anti-canonical divisor −KS is ample and that 2KS is Cartier. The log del Pezzo surfaces of index two are shown to be constructed from data (X,E,∆) called fundamental triplets consisting of a non-singular rational surface X, a simple normal crossin...

let $r$ be a commutative noetherian ring with non-zero identity and $fa$ an ideal of $r$. let $m$ be a finite $r$--module of finite projective dimension and $n$ an arbitrary finite $r$--module. we characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(m,n)$ in certain serre subcategories of the category of modules from upper bounds. we define and study the properti...

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ called an unbendable on $X$. Associated with it the variety of minimal tangents (VMRT) at point $x \in C$, which germ submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_...

It is a well-known result of Auslander and Reiten that contravariant finiteness the class P ∞ fin (of finitely generated modules finite projective dimension) over an Artin algebra sufficient condition for validity finitistic dimension conjectures. Motivated by fact dimensions can alternatively be computed Gorenstein dimension, we examine in this work counterpart Auslander–Reiten condition, name...

The main purpose of this article is to offer some characterizations of $delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:Let $n > 1$ be an integer and let $mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Si...

let $mathcal{a}$ be a $c^*$-algebra and $z(mathcal{a})$ the‎ ‎center of $mathcal{a}$‎. ‎a sequence ${l_{n}}_{n=0}^{infty}$ of‎ ‎linear mappings on $mathcal{a}$ with $l_{0}=i$‎, ‎where $i$ is the‎ ‎identity mapping‎ ‎on $mathcal{a}$‎, ‎is called a lie higher derivation if‎ ‎$l_{n}[x,y]=sum_{i+j=n} [l_{i}x,l_{j}y]$ for all $x,y in  ‎mathcal{a}$ and all $ngeqslant0$‎. ‎we show that‎ ‎${l_{n}}_{n...

We show that a properly stratified algebra is Gorenstein if and only the characteristic tilting module coincides with cotilting module. further algebras A enjoy strong homological properties such as all projective modules being endomorphism rings End ( Δ i ) Frobenius algebras. apply our results to study of are minimal Auslander-Gorenstein in sense Iyama-Solberg calculate under suitable conditi...