Direct summands of infinite-dimensional polynomial rings

Let $k$ be a field and $R$ a pure subring of the infinite-dimensional polynomial ring $k[X_1,\ldots ]$. 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 $k[X_1,\ldots ]$. This provides several examples of non Noetherian Cohen-Macaulay rings (e.g., Veronese, determinantal and Grassmanian rings).