Multiplicative structures on power series and the construction of skewfields (Q1125895)

Let \(R\) be a ring (associative, with 1); the author studies the multiplications on the additive group of formal power series \(R[[t]]\), which make it into a ring. If coefficients are written on the left, each such multiplication is completely determined by the value of \(ta\), for \(a\in R\), and thus corresponds to a sequence \(\underline M=(M_0,M_1,\dots)\) of endomorphisms of \(R^+\), where \(ta=\sum M_i(a)t^{i+1}\). Associative multiplications, here called twistors, are characterized by the condition \(t(ab)=(ta)b\). Examples are multiplications of the form \((1,D,D^2,\dots)\), where \(D\) is a derivation on \(R\). When \(R\) is a free algebra, twistors can be constructed by defining \(M_i\) recursively and for more general \(R\) this leads to a construction in terms of conditions on the defining ideal of \(R\). It turns out that \(R\) has a twistor with \(M_0=1\), other that the trivial one: \((1,0,0,\dots)\), if and only if \(R\) has a nonzero derivation. The additive group \(A\) of \(R[[t]]\) along with any multiplication \(\underline M\) admits an endomorphism \(\Lambda\underline M:f\to tft^{-1}\), here called lifting of \(\underline M\). These liftings are identified with endomorphisms of \(A\) fixing 1 and commuting with right multiplication by \(t\). Under this identification the convolution product of multiplications corresponds to the composition of endomorphisms. The author considers a number of other properties of multiplications such as the commutativity under the convolution product and examines the special case when \(R\) is a skew field. He shows that associativity of \(R((t))\) is determined by its effect on \(R[[t]]\) and ends with the well-known observation that in this case \(R((t))\) with an associative multiplication is again a skew field.