  The algebra $\mS_n$ of one-sided inverses of a polynomial algebra $P_n$ in
$n$ variables is obtained from $P_n$ by adding commuting, {\em left} (but not
two-sided) inverses of the canonical generators of the algebra $P_n$. The
algebra $\mS_n$ is a noncommutative, non-Noetherian algebra of classical Krull
dimension $2n$ and of global dimension $n$ which is not a domain. If the ground
field $K$ has characteristic zero then the algebra $\mS_n$ is canonically
isomorphic to the algebra $K< \frac{\der}{\der x_1}, ..., frac{\der}{\der x_n},
\int_1, ..., \int_n>$ of scalar integro-differential operators. %Ignoring
non-Noetherian % property, the algebra $\mS_n$ belongs to a family of algebras
% like the $n$th Weyl algebra $A_n$ and the polynomial algebra %$P_{2n}$. It is
proved that $ {\rm K}_1(\mS_n)\simeq K^*$. The main idea is to show that the
group $\GL_\infty (\mS_n)$ is generated by $K^*$, the group of elementary
matrices $E_\infty (\mS_n)$ and $(n-2)2^{n-1}+1$ explicit (tricky) matrices and
then to prove that all the matrices are elementary. For each nonzero idempotent
prime ideal $\gp$ of height $m$ of the algebra $\mS_n$, it is proved that $$
\rm K}_1(\mS_n, \gp)\simeq K^*, & \text{if}m=1, \Z^{\frac{m(m-1)}{2}}\times
K^{*m}& \text{if}m> 1.$$
