Derivations in free power series rings and free associative algebras

Abstract

A derivation d in any associative ring R is a linear mapping such that (ab)d = adb + abd, any a, b [set membership] R. The kernel of d is a subring of R which can sometimes be a ring of the same type as R. In particular, if R is a free power series ring, FX1,...,Xq>>, over a commutative field of characteristic zero, we find conditions under which Ker d is again a free power series ring. This happens e.g. if all the nonzero elements of the set {Xid; 1 = 1,...,q} are homogeneous of the same order, or if at least one element in this set has a nonzero constant term. For every derivation d in a complete inversely filtered F-algebra S satisfying the [n-term] inverse weak algorithm it is at least true that Ker d is [an n-fir] a semifir, i.e. Ker d is then again a ring in which every finitely generated [by at most n generators] right ideal is a free right S-module of unique rank. This is also true for the fixed rings of suitably chosen automorphisms of S, for if [alpha] is an automorphism which maps every element onto itself plus an element of higher order, then log [alpha] is a derivation such that Fix = Ker (log [alpha]). In a free associative algebra F, X a countable set, the kernel of any derivation d such that the nonzero elements of the set {xd; x [set membership] X} are homogeneous of the same degree, is also a free associative algebra over F. In particular, the kernel of the derivation d/dx has a free generating set consisting of {y [set membership] x; y=/ x} together with the set of all commutators of the form [..[[y,x],..,x]. This makes it possible to regard F(X) as a skew polynomial ring in x over Ker d/dx, a fact which characterizes x up to a "constant" in Ker d/dx .