Noetherian properties in composite generalized power series rings

Let (Gamma, <=) be a strictly ordered monoid, and let Gamma* = Gamma\{0}. Let D subset of E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set D + [[E-Gamma*(,) (<=)]] = {f is an element of [[E-Gamma*(,) (<=)]] vertical bar f (0) is an element of D} and D + [[I-Gamma*(,) (<=)]] = {f is an element of [[D-Gamma*(,) (<=)]] vertical bar f (alpha) is an element of I, for all alpha is an element of Gamma*}. In this paper, we give necessary conditions for the rings D + [[E-Gamma*(,) (<=)]] to be Noetherian when (Gamma, <=) is positively ordered, and sufficient conditions for the rings D + [[E-Gamma*(,) (<=)]] to be Noetherian when (Gamma, <=) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D + [[I-Gamma*(,) (<=)]] to be Noetherian when (Gamma*, <=) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D + (X-1,..., X-n) E [X-1,..., X-n] and D + (X-1,..., X-n) I [X-1,..., X-n] to be Noetherian.