  An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$
of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an
arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra
$\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a
domain. The proof is based on a generalization of the Theorem of Kaplansky (on
the projective dimension) obtained in the paper. As a consequence it is proved
that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A)
+n.$$
