 Noncommutative formal power series ring

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.