On Cohen–Macaulayness and depth of ideals in invariant rings

We investigate the presence of Cohen–Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a p-group is not Cohen–Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen–Macaulay factorial rings cannot contain Cohen–Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen–Macaulay, extending a result of Fleischmann.