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单词 InvertibleFormalPowerSeries
释义

invertible formal power series


Theorem.  Let R be a commutative ring with non-zero unity.  A formal power series

f(X):=i=0aiXi(1)

is invertiblePlanetmathPlanetmath in the ring R[[X]]  iff  a0 is invertible in the ring R.

Proof.1.  Let f(X) have the multiplicative inverseMathworldPlanetmathg(X):=i=0biXi.  Since

f(X)g(X)=i=0j=0iajbi-jXi= 1,

we see that  a0b0=1, i.e. a0 is an invertible element (unit) of R.

2.  Assume conversely that a0 is invertible in R.  For making from a formal power series

g(X):=i=0biXi(2)

the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of f(X)=i=0aiXi, we first choose  b0:=a0-1.  For all already defined coefficients b0,b1,,bi-1 let the next coefficient be defined as

bi:=-a0-1(a1bi-1+a2bi-2++aib0).

This equation means that

j=0iajbi-j=a0bi+a1bi-1+a2bi-2++aib0

vanishes for all  i=1, 2,;  since  a0b0=1,  the productPlanetmathPlanetmath of the formal power series (1) and (2) becomes simply equal to 1.  Accordingly, f(x) is invertible.

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更新时间:2025/5/4 16:27:22