TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A zero-constant formal series has a compositional inverse exactly when its linear coefficient is a unit

Statement

Let $R$ be a commutative ring and $f \in x R ⟦ x ⟧$ . There is a unique $g \in x R ⟦ x ⟧$ such that

$f \circ g = x = g \circ f$

if and only if $[x] f$ is a unit in $R$ . In the zero ring the assertion holds with the unique zero series; when $R$ is nonzero, a zero linear coefficient cannot satisfy the criterion.

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

If $g$ and $h$ both have zero constant coefficient then $(f \circ g) \circ h = f \circ (g \circ h)$ ; also $f \circ x = f$ and $x \circ f = f$ (Substitution by a zero-constant series is a ring homomorphism, and composition is associative when both inner series have zero constant coefficient).

Proof

technique · recursive coefficient construction

1.1

If $f \circ g = x$ , then the coefficient of $x$ is $[x] f [x] g = 1$ . Thus $[x] f$ is a unit. The same equation also determines $[x] g$ as its inverse.

given

1.2

Conversely write $f = a_{1} x + a_{2} x^{2} + \dots$ with $a_{1}$ a unit. Choose $b_{1} = a_{1}^{- 1}$ . After $b_{1}, \dots, b_{n - 1}$ have been chosen, the coefficient of $x^{n}$ in $f \circ (b_{1} x + \dots + b_{n} x^{n})$ is $a_{1} b_{n} + c_{n}$ , where $c_{n}$ depends only on the earlier $b_{j}$ . Set $b_{n} = - a_{1}^{- 1} c_{n}$ . The resulting $g$ has $f \circ g = x$ , and the same equations show that it is the unique left inverse.

given

2.1

Apply the construction to $g$ : its linear coefficient $a_{1}^{- 1}$ is a unit, so there is $h$ with $g \circ h = x$ . Associativity gives $f = f \circ x = f \circ (g \circ h) = (f \circ g) \circ h = x \circ h = h$ . Thus $g \circ f = x$ as well, and any two-sided inverse is the already unique solution of $f \circ g = x$ .

step 1.2givenF1

3.1

In the zero ring, $x = 0$ and the sole series is its own inverse. In a nonzero ring, $0$ is not a unit, so a zero linear coefficient fails necessity. Together with steps 1.1-2.1 this proves the equivalence and uniqueness.

step 1.1step 1.2step 2.1given∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)
Herbert S. Wilf, generatingfunctionology (standard reference, not scraped)