Noncommutative formal power series ring ($k⟨⟨x_1,\dots ,x_r⟩⟩$)

For a field $k$ and finitely many generators, this ring consists of arbitrary coefficient families indexed by finite words in the generators. Addition is coefficientwise and multiplication uses concatenation: the coefficient of $w$ in $FG$ is ${\sum }_{uv=w}{F}_u{G}_v$, a finite sum. Finite series form the free associative algebra. If $F,G\ne 0$, their lowest nonzero homogeneous parts multiply to a nonzero polynomial, since a word of fixed length has a unique splitting at a specified position. Thus the ring is a noncommutative domain.

A noncommutative formal power series over a field is a unit exactly when its constant term is nonzero. Necessity follows because the constant term is a ring homomorphism to the coefficient field. For sufficiency, write $F=c(1+h)$ with $c\ne 0$ and $h$ of zero constant term. The formal geometric series ${\sum }_{j\ge 0}(-h{)}^j$ is well-defined because only $j\le d$ can contribute to a word of length $d$. Its left and right products with $1+h$ are $1$, by the finite geometric-series identity in each fixed degree. Multiplying by $c^{-1}$ gives the inverse of $F$.

A noncommutative formal power series is an element of a noncommutative formal power series ring. Its constant term is the coefficient of the empty word. Formal sums are interpreted coefficient by coefficient; analytic convergence is not required.