Abstract
We prove various characterisations for the Cohen-Macaulay property for the invariant ringk[x(1), horizontal ellipsis , x(n)](G), wherekis a PID.We also show that, except for one case, all the invariant rings DOUBLE-STRUCK CAPITAL Z[x(1), x(2)](G)and DOUBLE-STRUCK CAPITAL Z[x(1), x(2), x(3)](G)are Cohen-Macaulay. We also provide some sufficient conditions to obtain a polynomial invariant ring over the ring of integers.