the chain condition often is "inherited" by sub-objects. for example, all subspaces of a noetherian space, are noetherian themselves; all subgroups and quotient groups of a noetherian group are likewise, noetherian; and, "mutatis mutandis", the same holds for submodules and quotient modules of a noetherian module. all quotient rings of a noetherian ring are noetherian, but that does not necessarily hold for its subrings. the chain condition also may be inherited by combinations or extensions of a noetherian object. for example, finite direct sums of noetherian rings are noetherian, as is the ring of formal power series over a noetherian ring.