  Let k be a field and R a pure subring of the infinite-dimensional polynomial
ring k[X1;...]. If R is generated by monomials, then we show that the equality
of height and grade holds for all ideals of R. Also, we show R satisfies the
weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay
property of the invariant ring of the action of a linearly reductive group
acting by k-automorphism on k[X1;...]. This provides several examples of
non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and
Grassmanian rings).
