On divided commutative semigroups (Q2747345)

Let \(S\) be a commutative cancellative torsionfree additive monoid. A prime ideal \(P\) of \(S\) is said to be divided if \(P\) is comparable to every principal ideal of \(S\). If every prime ideal \(S\) is divided, then \(S\) is called divided semigroup. In this paper, the author gives the corresponding semigroup versions of the ring-theoretic results for divided commutative rings given by \textit{A. Badawi} [in Commun. Algebra 27, No. 3, 1465-1474 (1999; Zbl 0923.13001)]. For example, the author shows that if \(S\) is an atomic divided semigroup, then \(S\) has Krull dimension \(\leq 1\), and that a finitely generated prime ideal of a divided semigroup is maximal.