On extending higher derivations generated by cup products to the integral closure

Derivations and the Integral Closure of Ideals

Let (R, m) be a complete local domain containing the rationals. Then there exists an integer l such that for any ideal I ⊆ R, if f ∈ m, f / ∈ In, then there exists a derivation δ of R with δ(f) / ∈ In+l. This note grew out of a question raised by C. Huneke some years ago: Suppose (R, m) is a complete local domain with perfect residue class field k which it contains. Call an element x ∈ R deriva...

Extending Higher Derivations to Rings and Modules of Quotients

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsio...

Local higher derivations on C*-algebras are higher derivations

Let $mathfrak{A}$ be a Banach algebra. We say that a sequence ${D_n}_{n=0}^infty$ of continuous operators form $mathfrak{A}$ into $mathfrak{A}$ is a textit{local higher derivation} if to each $ainmathfrak{A}$ there corresponds a continuous higher derivation ${d_{a,n}}_{n=0}^infty$ such that $D_n(a)=d_{a,n}(a)$ for each non-negative integer $n$. We show that if $mathfrak{A}$ is a $C^*$-algebra t...

Prime Higher Derivations on Algebras

on the teacher-generated v.s. leaner-generated noticing-the-gap activities in language classes

abstract the purpose of this study is twofold: on the one hand, it is intended to see what kind of noticing-the –gap activity (teacher generated vs. learner generated) is more efficient in teaching l2 grammar in classroom language learning. on the other hand, it is an attempt to determine which approach of the noticing-the-gap- activity is more effective in the long- term retention of grammar...