We extend a lemma by Matsuda about the irreducibility of the binomial X π − 1 in the semigroup ring F [X; G], where F is a field, G is an abelian torsion-free group and π is an element of G of height (0, 0, 0, . . . ).In our extension, G is replaced by any submonoid of (Q + , +). The field F , however, has to be of characteristic 0. We give an application of our main result.

In 2008 N. Q. Chinh and P. H. Nam characterized principal ideal domains as integral domains that satisfy the following two conditions: (i) they are unique factorization domains, and (ii) all maximal ideals in them are principal. We improve their result by giving a characterization in which each of these two conditions is weakened. At the same time we improve a theorem by P. M. Cohn which characterizes principal ideal domains as atomic Bézout domains. We will also show that every PC domain is AP and that the notion of PC domains is incomparable with the notion of pre-Schreier domains (hence with the notions of Schreier and GCD domains as well).2010 Mathematics Subject Classification. Primary 13F15; Secondary 13A05, 13F10.

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We characterize the Puiseux monoids M for which the irreducible and the prime elements in the monoid ring F [X; M ], where F is a field, coincide. We present a diagram of implications between some types of Puiseux monoids, with a precise position of the monoids M with this property.

We investigate the atomicity and the AP property of the semigroup rings F [X; M ], where F is a field, X is a variable and M is a submonoid of the additive monoid of nonnegative rational numbers. The main notion that we introduce for the purpose of the investigation is the notion of essential generators of M . (2010): 13F15, 13A05 Mathematics Subject Classification