Multiplicative Structures on Power Series and the Construction of Skewfields

✍ Scribed by Paulo Ribenboim

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 297 KB
• Volume: 185
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0325

✦ Synopsis

Let R be an associative ring with unit element, let R t be the additive group of formal power series in one indeterminate with coefficients in R.

ww xx In this paper, we determine all possible multiplications on R t , which are distributive and satisfy other reasonable conditions.

ww xx To each multiplication on R t we associate an element of E E, the set Ž . of sequences M s M , M , M , . . . , where each M is an endomorphism

canonical correspondence is a bijection. The sequences corresponding to associative multiplications are fully characterized and are called the twistors of R. We give a method for constructing twistors and we show that sequences of iterated derivations of R are twistors, but not every twistor is of this type. Moreover, we show that there is a non-trivial twistor if and only if R has a non-zero derivation. ww xx Twistors may be lifted to endomorphisms of the additive group of R t . We determine the behaviours of the lifting with respect to multiplications.