Corrigendum to “Krull property of generalized power series rings”

초록

There is a wrong argument in the last part of the proof of [2, Theorem 3.7]: The quotient field of [Formula presented] say [Formula presented] as a subring of [Formula presented]and the quotient field of [Formula presented] say [Formula presented] as a subring of the quotient field of [Formula presented]are different as sets. The intersection [Formula presented]holds as subsets of T, but the intersection [Formula presented]as subsets of the quotient field of [Formula presented]does not hold. We give a simple and correct proof of [2, Theorem 3.7], which is similar to the proof of [2, Theorem 3.4]. Theorem 1 Let [Formula presented]be a torsion-free, ≤-cancellative, subtotally ordered monoid, and let K be a field. If V is a discrete valuation monoid, then [Formula presented]is a Krull domain. Proof Let G be the quotient group of V. Since [Formula presented]is ≤-cancellative, the order ≤ on V extends naturally to a compatible order, still denoted by ≤, on G ([2, Proposition 2.3(6)]). Then [Formula presented]is a torsion-free subtotally ordered group ([2, Proposition 2.3(7)]). Since [Formula presented]is a torsion-free group (and hence cancellative), there exists a compatible strict total order [Formula presented]on G ([2, Proposition 2.4]). Then we have [Formula presented]([2, Lemma 3.3]). Moreover, it is equal to [Formula presented] since [Formula presented]is a field ([2, Proposition 2.5(2)]). Assume that V is a discrete valuation monoid. Then [Formula presented] where H is the group of units of V and [Formula presented]is the submonoid of [Formula presented]of nonnegative integers (cf. [1, Theorem 6.8]). Therefore, by [3, (5.5)], [Formula presented]is a Noetherian ring. Also, since [Formula presented]is a totally ordered and completely integrally closed monoid, [Formula presented]is a completely integrally closed domain by [4, (5.3)]. Thus [Formula presented]is an integrally closed Noetherian domain and hence a Krull domain. Therefore, it follows that [Formula presented]is also a Krull domain. □ © 2023 Elsevier B.V.

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