(1) In 1963, Bass proved that any nonfinitely generated projective module is free over a connected Noether ring.
(2) Quillen–Suslin theorem states that any finitely generated projective module over a polynomial ring with finite variables is free.
By (1) and (2), we can see that any projective module over a polynomial ring with finite variables is free. So it is natural to ask what is about infinite variables' polynomial ring, i.e., are projective modules free over a polynomial ring with infinite indeterminates over a field? Furthermore, how to judge if a module is a projective module over a polynomial ring with infinite indeterminates over a field?
For example, let $k$ be a field and let $k[x_1,...,x_n,...]$ be the polynomial ring in countably many indeterminate over the field $k$, is the ideal $(x_1,...,x_n,...)$ projective module over $k[x_1,...,x_n,...]$?