Irreducibility of Certain Binomials in Semigroup Rings for Nonnegative Rational Monoids

Abstract

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.

For Which Puiseux Monoids Are Their Monoid Rings Over Fields Ap?

“…The next theorem is the main result of our paper [5]. It is not true for fields of positive characteristics (as the examples and related questions in [5] illustrate).…”

“…The next theorem is the main result of our paper [5]. It is not true for fields of positive characteristics (as the examples and related questions in [5] illustrate). We use this theorem in the proof of the main theorem of this paper (namely Theorem 4.3) and that explains why we need in our main theorem the assumption that F is of characteristic 0.…”

“…We begin by recalling some definitions and facts. All the notions that we use but do not define in this paper, as well as the definitions and results for which we do not specify the source, can be found in the classical reference books [8] by P. M. Cohn, [18] and [19] by R. Gilmer, [32] by I. Kaplansky, and [35] by D. G. Northcott, as well as in our papers [5] and [22]. We also recommend the paper [1] in which the work of R. Gilmer is nicely presented, in particular his work on characterizing cancellative torsion-free monoids M for which the monoid domain F [X; M ] has the property P for various properties P .…”

“…of p-heights of a as p goes through all prime numbers in the increasing order is called the height sequence of a. In our paper [5] we considered the elements of height (0, 0, 0, . .…”

“…Note that the additive submonoids of Q + have received a lot of attention by the researchers in the areas of commutative semigroup theory and factorization theory during the last few years (see, for example, [4,5,9,12,22,23,25,26,32]) and they even got a special name, Puiseux monoids, so we will be using that name from now on. Of course, Puiseux monoids were also used before; the paper [27] is one of the well-known instances.…”