Composite Hurwitz Rings Satisfying the Ascending Chain Condition on Principal Ideals

Let D ⊆ E be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D,E) and H(D, I) (resp., h(D,E) and h(D, I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D,E) satisfies the ascending chain condition on principal ideals if and only if h(D,E) satisfies the ascending chain condition on principal ideals, if and only if ∩ n≥1 a1 · · · anE = (0) for each infinite sequence (an)n≥1 consisting of nonzero nonunits of D. We also prove that H(D, I) satisfies the ascending chain condition on principal ideals if and only if h(D, I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.