  Refining an idea of Rosenmann and Rosset we show that the now widely studied
classical Leavitt algebra $L_K(1,n)$ over a field $K$ is a ring of right
quotients of the unital free associative algebra of rank $n$ with respect to
the perfect Gabriel topology defined by powers of an ideal of codimension 1,
providing a conceptual, variable-free description of $L_K(1, n)$. This result
puts Leavitt (path) algebras on the frontier of important research areas in
localization theory, free ideal rings and their automorphism groups, quiver
algebras and graph operator algebras. As applications one obtains a short,
transparent proof for the module type $(1,n)\, (n\geq 2)$ of Leavitt algebra
$L_K(1, n)\, (n\geq 2)$, and the fact that Leavitt path algebras of finite
graphs are rings of quotients of corresponding ordinary quiver algebras with
respect to the perfect Gabriel topology defined by powers of the ideal
generated by all arrows and sinks. In particular, the Jacobson algebra of
one-sided inverses, that is, the Toeplitz algebra, can also be realized as a
flat ring of quotients, further illuminating the rich structure of these
beautiful, useful algebras.
