THE ASCENDING CHAIN CONDITION ON PRINCIPAL IDEALS IN COMPOSITE GENERALIZED POWER SERIES RINGS

Let D (subset of) under bar E be an extension of commutative rings with identity, I a nonzero proper ideal of D, (Gamma, <=) a strictly totally ordered monoid such that 0 <= alpha for all alpha is an element of Gamma, and Gamma* = Gamma \ {0}. Let 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 some conditions for the rings D + [I-Gamma*, (<=)] and D + [I-Gamma*,( <=)] to satisfy the ascending chain condition on principal ideals.