= Noncommutative formal power series ring
{title2=$k\langle\!\langle x_1,\ldots,x_r\rangle\!\rangle$}
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 $\sum_{uv=w}F_uG_v$, a finite sum. Finite series form the <free associative algebra>. If $F,G\ne0$, 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>.
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