Catalan paths and quasi-symmetric functions

We investigate the quotient ring of the ring of formal power series over the closure of the ideal generated by non-constant quasi-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 and above the line . We investigate as well the quotient ring of polynomial ring in variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of is bounded above by the th Catalan number.