  We investigate the quotient ring $R$ of the ring of formal power series
$\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant
quasi-\break symmetric functions. We show that a Hilbert basis of the quotient
is naturally indexed by Catalan paths (infinite Dyck paths). We also give a
filtration of ideals related to Catalan paths from $(0,0)$ and above the line
$y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in
$n$ variables over the ideal generated by non-constant quasi-symmetric
polynomials. We show that the dimension of $R_n$ is bounded above by the $n$th
Catalan number.
