FOR WHICH PUISEUX MONOIDS ARE THEIR MONOID RINGS OVER FIELDS AP?

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. Mathematics Subject Classification (2010): 13F15, 13A05, 20M25, 20M13, 13Gxx, 20M14


Introduction
If M is a commutative monoid, written additively, and F is a field, the monoid ring F [X; M ] consists of the polynomial expressions (also called polynomials) f = a 0 X α0 + a 1 X α1 + · · · + a n X αn , where n ≥ 0, a i ∈ F , and α i ∈ M (i = 0, 1, . . . , n). F [X; M ] is an integral domain if and only if M is cancellative and torsion-free by [19,Theorem 8.1]. Since we will be dealing exclusively with cancellative torsion-free monoids, we will call monoid rings the element X 1/5 is an atom which is not prime (see [22]). So we naturally come to the following question: for which submonoids M of Q + is the monoid domain F [X; M ] AP? The goal of this paper is to give an answer to this question. In addition to that, we will present an implication diagram between various properties of submonoids of Q + in which we will precisely position the monoids M for which F [X; M ] is AP.
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. As we said, in this current paper we will be mainly

Notation and preliminaries
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 .
We use ⊆ to denote inclusion and ⊂ to denote strict inclusion. We denote All the monoids used in the paper are assumed to be commutative and written additively. Thus a monoid is a nonempty set M with an associative and commu- and differences between the ideal theories of monoids and integral domains are studied, for example, in the classical references [3,29,31], as well as in the recent papers [15,16,17].) The notion of a Prüfer monoid was introduced in [21, p. 223-224] (see also [19, p. 166-167]). We include the possibility that M = {0}.
Definition 2.1. We say that a monoid M is a Prüfer monoid it it is a union of an increasing sequence of cyclic submonoids.
We now give some observations and definitions in the context of Puiseux monoids.
If M is a Puiseux monoid and A a generating set of M , note that: (c) if a 1 , . . . , a n ∈ A are not atoms of M , then A = A \ {a 1 , . . . , a n } is also a generating set of M .
Definition 2.2. We say that a Puiseux monoid M is: In this paper all rings are integral domains, i.e., commutative rings with identity in which xy = 0 implies x = 0 or y = 0. A non-zero non-unit element x of an integral domain R is said to be irreducible (and called an atom) if x = yz with y, z ∈ R implies that y or z is a unit. A non-zero non-unit element x of an integral prime element is an atom, but not necessarily vice-versa. Two elements x, y ∈ R are said to be associates if x = uy, where u is a unit. We then write x ∼ y.
We now give the definitions of some kinds of integral domains. The definitions of all other kinds that we use in the paper can be found in [19] and/or in the references given in the proof of Corollary 4.4 below.     It is easy to see that an equivalent definition of the gcd/lcm condition is that for any two elements m 1 n 1 , m 2 n 2 ∈ M , written in reduced form, at least one of which is The next proposition is a generalization of the theorem of Daileda, mentioned in Introduction. (The proof that we gave in [22] follows Daileda's proof of his theorem from [12].) The proposition is a step toward the main theorem of the paper, namely     In the next theorem we give several equivalent conditions for Puiseux monoids.
Since gcd(x 1 y 2 , x 2 y 1 ) = 1, there are k, l ∈ N 0 such that We will assume that this difference is equal to d, the reasoning being similar if it is equal to −d. Since Hence, since M is difference closed, i.e., dkx 2 y 1 − dlx 1 y 2 ey 1 y 2 ∈ M.
Thus M satisfies the gcd/lcm property.
Hence M contains the element Thus M is difference closed.   We can now prove the main theorem of the paper. It characterizes the Puiseux monoids for which F [X; M ] is an AP domain when F is a field of characteristic 0. n > 1. We will show that then the element X π − 1 of F [X; M ] is irreducible, but not prime. By Theorem 3.7, X π − 1 is irreducible in F [X; M ]. We now show that it is not prime. We have π = mp + n pn . Hence We have: Then where α 1 > α 2 > · · · > α k−1 > 0 and g 2 , . . . , g k−1 ∈ F . It follows that Note that since, otherwise, we would get (i−2)n = mp, which is not possible since gcd(m, n) = 1 and gcd(p, n) = 1. Note also that It follows that the exponent mp + n − (r + 1) p from the left-hand side cannot be obtained from X mp+n pn · (X α1 + g 2 X α2 + · · · + g k−1 X α k−1 − 1) since r was the largest integer for which (9) holds, nor from since (11) holds and no α i with i < r can be equal to any mp + n − i p , i = r + 1, r + 2, . . . , mp + n − 1. We got a contradiction, hence (2) does not hold. So X π − 1 is not prime.
The theorem is proved.   (2) In Question 5.7 from our paper [22]   and relations between them are analyzed in detail in [2]).