Direct summands of infinite-dimensional polynomial rings

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).