On 2-dimensional Noetherian semigroups and a principal ideal theorem

Let D be a Noetlerian integral domain with the integral closure D, and K the quotient field of D. The Krull-Akizuki theorem states that , if dim (D) =1, then any ring between D and K is Noetherian and its dimension is at most 1. The Mori-Nagata theorem states that D is a Krull ring for any Noetherian domain D. Moreover, Nagata proved that, if D is of dimension 2 , then D is Noetherian (cf. [N, (33.12) Theorem). In [M1] we proved the Krull-Akizuki theorem for semigroups. In [M2] we proved the Mori-Nagata theorem for semigroups . The aims of this paper are to prove the following Theorem and to answer to the following question. THEOREM. Let S be a 2-dimensional Noetherian semigroup . Then the integral closure S of S is a Noetherian semigroup.