Inverse in noncommutative power series ring

Let $k$ be a field and let $X_1, X_2, \ldots , X_n$ be formal noncommuting variables and let $K\langle \langle X_1, X_2, \ldots , X_n\rangle \rangle$ be the formal noncommutative power series ring in these variables.

Suppose $F$ is a power series in this ring such that its constant term is nonzero. How to prove that $F^{-1}$ exists?

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$k=K$?${}{}{}{}$ – Pierre-Yves Gaillard Nov 29 '11 at 15:48

3 Answers

on page 5 in his paper "POWER SERIES OVER THE GROUP RING OF A FREE GROUP AND APPLICATIONS TO NOVIKOV-SHUBIN INVARIANTS" Roman Sauer answers your Question, I think,

AB, martini.

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It works like in the commutative case:

We can assume $F=1-G$, where the constant term of $G$ is $0$. Then the sum $S$ of the nonnegative powers of $G$ converges in the obvious topology, and we have $FS=1$.

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Let's try it for $n=1$.

You can then define $F^{-1}$ by induction.

Write $F=a_0 +a_1 X+\ldots$. Write $F^{-1}= b_0 + b_1 X+\ldots$, with

$b_0 = a_0^{-1}$,

$a_0 b_1 + a_1 b_0 =0$,

$a_0 b_2 + a_1 b_1 + a_2 b_0 =0$,

etc.

From this you can guess a formula for $b_i$ and prove its correctness (by induction), i.e., verify that $F^{-1} F = 1$.

For $n>1$, I haven't tried writing it out but it shouldn't be more difficult.

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I was worried about noncommutativity. In the n=1 case, noncommutativity makes no difference whatsoever. So please address the case of higher n's. – Valeriya Sep 30 '11 at 13:25