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The free associative algebra has a basis of finite words in its generators, including the empty word as identity. Multiplication concatenates words without commutativity relations. It is the path algebra of the quiver with r loops. With one generator it is a polynomial algebra.

There is one constant path and $r$ loop generators; every other path is a word in those loops. Distinct words are distinct basis paths, so there are no relations among the generators. Thereforethe free associative algebra on $r$ generators, with each loop sent to its corresponding generator. For $r=1$ this is $k[x_1]$; for $r\ge 2$ it is noncommutative. The quiver with one loop is the $r=1$ case of the quiver with r loops.