Skew power series rings and derivations
The authors study skew power series rings in z over a skew field F with the commutation rule \(az=za^{\delta_ 0}+z^ 2a^{\delta_ 1}+...,\) where multiplication preserves the order, thus \(a^{\delta_ 0}=0\) implies \(a=0\). The appropriate sequence \((\delta_ 0,\delta_ 1,...)\) (here called ''admissible'', but also ''higher \(\delta_ 0\)-derivations'') consists of additive mappings of F satisfying an analogue of Leibniz's formula. For a commutative field F of characteristic 0 it is shown that in any admissible sequence \((1,\delta_ 1,\delta_ 2,...)\) the terms \(\delta_ 1,...,\delta_{n-1}\) determine \(\delta_ n\) up to a derivation of F. A particular case is that where the extra term is \(a_ n\delta\) for \(a_ n\in F\) and this leads to polynomials \(\delta_ n=g_ n(\delta)\) whose coefficients depend on \(a_ 1,...,a_ n\). For any a's in the kernel of \(\delta\) this leads to an admissible sequence \((1,g_ 1(\delta),g_ 2(\delta),...),\) but it is not known whether this is admissible for any choice of a's in F. It is shown that for any derivation \(\delta\) and any integer \(k\geq 1\), the definition \(\delta_ i=0\) for \(k\nmid j\), \(\delta_{nk}=(1/n!)(1+k)(1+2k)...(1+(n- 1)k)\delta^ k\) yields an admissible sequence.
scientific article; zbMATH DE number 3861383