A characterization of weakly Krull monoid algebras

Let $D$ be a domain and let $S$ be a torsion-free monoid whose quotient group satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra $D[S]$ is weakly Krull. As corollaries, we obtain the results on when $D[S]$ is Krull resp. generalized Krull, due to Chouinard resp. El Baghdadi and Kim. Furthermore, we deduce Chang's theorem on weakly factorial monoid algebras and we characterize the weakly Krull domains among the affine monoid algebras.


Introduction
When considering monoid algebras D [S], an immediate question is whether certain properties of the domain D and the monoid S carry over to the monoid algebra and conversely. A lot of such properties are studied in the textbook by Gilmer [15], among them the property of being a Krull domain. For monoid algebras, this property is completely characterized in terms of the domain and the monoid, and this characterization was originally proved by Chouinard [9]. There he was the first to give the definition of a Krull monoid, which in turn was preparing the ground for a whole new research area based on the fact that a domain is Krull if and only if its multiplicative monoid is a Krull monoid. For a detailed study of Krull monoids, see [14]. Many well-studied classes of domains (e.g. non-principal orders in number fields) fail to be Krull, but still share many important properties with Krull domains. To investigate them, two generalizations of Krull domains and monoids were introduced; namely generalized Krull and weakly Krull domains and monoids. The latter were first studied in [3], although not named weakly Krull domains there. It still holds true, that a domain is generalized (resp. weakly) Krull if and only if its multiplicative monoid is generalized (resp. weakly) Krull. A profound introduction to these classes of monoids can be found in [20]. A divisor-theoretic characterization of weakly Krull monoids was first given by Halter-Koch in [19], where one can also find a proof of the statement that a domain is weakly Krull if and only if its multiplicative monoid is. In 2009, Chang [7] gave a characterization of when the monoid algebra is weakly factorial (i.e. every non-zero non-unit is a product of primary elements) under the assumption that the quotient group of the monoid satisfies the ACC (ascending chain condition) on cylic subgroups. Namely, D[S] is weakly factorial if and only if D is a weakly factorial GCD-domain and S is a weakly factorial GCD-monoid. In this paper, he also posed the question of when a monoid algebra is weakly Krull (note that a domain is weakly factorial if and only if it is weakly Krull with trivial t-class group). In 2016, El Baghdadi and Kim [12] gave a complete characterization of when a monoid algebra is generalized Krull (namely if and only if both, the domain and the monoid are generalized Krull and the quotient group of the monoid satisfies the ACC on cyclic subgroups). An immediate consequence of the results by Chouinard, and El Baghdadi and Kim is the fact that the polynomial ring over a (generalized) Krull domain is (generalized) Krull and one would hope so as well for the case of weakly Krull domains. That this is not the case was proven by D.D. Anderson, Houston and Zafrullah in [2,Proposition 4.11]. In fact, they showed that this is the case precisely for weakly Krull UMT-domains. Recall that a domain is said to be a UMT-domain if all prime ideals of the polynomial ring D[X] lying over (0) are maximal t-ideals. Answering Chang's question and trying to close the last gap for a complete picture of the generalizations of Krull domains, in Section 3, we give a characterization of when a monoid algebra is weakly Krull under the assumption that the quotient group of the monoid satisfies the ACC on cyclic subgroups. As a proof of the applicability of our description, in Section 4, we characterize the weakly Krull domains among the affine monoid algebras, which play an important role in the study of polytopes (see [6]). Moreover, we reobtain the results by Chouinard, El Baghdadi and Kim, and Chang.

Preliminaries
We assume some familiarity with monoid algebras and ideal systems. We consider monoids to be cancellative, commutative and unitary semigroups and write them additively. For a domain D and a monoid S, let D[S] denote the monoid algebra of D over S. It is well known that the monoid algebra D[S] is a domain if and only if the monoid S is torsion-free [15,Theorem 8.1] and in that case S amits a total order < compatible with its semigroup operation. Therefore we can write every element f ∈ D [S] in the form f = n i=1 d i X si with d i ∈ D, s i ∈ S and s 1 < s 2 < . . . < s n . For subsets P ⊆ D and H ⊆ S, we denote by P . A group satisfies the ACC (ascending chain condition) on cyclic subgroups if and only if it is of type (0, 0, . . .). For a proof of this statement and for further equivalent conditions, see [15, §14]. Let S be a monoid. We denote by • q(S) the quotient group of S, I is called a maximal t-ideal, if I is maximal among all proper t-ideals of S (and is therefore necessarily prime). S is said to be of t-dimension 1 (t-dim(S) = 1) if each maximal t-ideal of S is a minimal nonempty prime ideal. An analogous concept exists for domains and the reader is referred to [16] for the domain case and to [20] for the monoid case. We denote by X(D) (resp. X(S)) the height-one spectrum of D (resp. the minimal non-empty prime ideals of S) and will often also call elements of X(S) height-one prime ideals. A domain (resp. monoid) D is called weakly Krull if D = P ∈X(D) D P and X(D) is of finite character, meaning that every non-zero non-unit (resp. non-unit) is contained only in a finite number of height-one prime ideals (resp. minimal non-empty prime ideals). A weakly Krull domain (resp. monoid) D is called generalized Krull if D P is a valuation domain (resp. monoid) for all P ∈ X(D). For examples of generalized Krull monoids that do not stem from domains, we refer to [5] and [8].

Main Result
We start with an investigation of the height-one spectrum of monoid algebras, which will lead us to the definition of the central property characterizing weakly Krull monoid algebras. b j X tj with s 1 < . . . < s n and t 1 < . .
Since s u and t v were chosen minimal with respect to not lying in P we obtain s u < s u ′ and and we can again use the prime ideal correspondence.
In fact, the above proof shows us, that the primes coming from D[G] resp. K[S] are always height-one again. Thus, the only primes that can produce a strict inclusion are height-one primes from D resp. S that do not induce height-one primes in the monoid algebra.
Definition 3.2. Let D be a domain and let S be a torsion-free monoid. We say that The next remark is just a consequence of the previous definition and remark. Lemma 3.4. Let D be a domain with quotient field K and let S be a torsion-free monoid with quotient group G. Then the following hold: We now characterize the weakly Krull group algebras.
) not containing constants. Therefore in total we obtain that . To see that the intersection is of finite character, just note that an f ∈ D[G] can only be in finitely many height-one primes that come from K[G] since it is a factorial domain and that it also can only be in finitely many height-one primes coming from D since D is weakly Krull.
Next we characterize the weakly Krull monoid algebras over fields.
Proposition 3.6. Let K be a field and let S be a torsion-free monoid with quotient group G such that G satisfies the ACC on cyclic subgroups. Then K[S] is weakly Krull if and only if S is weakly Krull and K-UMT.
Proof. "⇒" Let K[S] be weakly Krull and P ∈ X(S). We have to show that K[P ] ∈ X(K[S]). We already know by the claim in the proof of Lemma 3.1 that K[P ] is a prime ideal, so it remains to prove its height is one. Let α ∈ P , then we obtain a primary decomposition and by P ∈ X(S) equality follows, hence K[P ] ∈ X(K[S]). To prove that S is weakly Krull, let α − β ∈ P ∈X(S) S P . Then for all P ∈ X(S) there exist α P ∈ S and β P ∈ S \ P such that α − β = α P − β P . Since for all a ∈ S we have a ∈ P if and only if a ∈ K[P ], it follows that X α−β = X αP −βP ∈ K[S] K[P ] . Clearly, β is in no height-one prime coming from K[G], so in total we obtain α − β ∈ G ∩ P ∈X Since P is a maximal t-ideal of S, by the choice of f = l i=1 a i X si (as always such that s 1 < . . . < s l ) we have for all i ∈ [1, l] that (P, s i ) t = S, because t-dim(S) = 1. Thus for all i ∈ [1, l] we can choose b One inclusion is trivial, so let h ∈ (X b1 , . . . , X bm , f ) −1 and note first, that h ∈ K[G] since hX b1 ∈ K[S]. Now write h = r k=1 c k X d k with d k ∈ G and d 1 < . . . < d r and we proceed by induction on r to show that h ∈ K[S]: If r = 1 this is clear, since for h = X d we have n1 , s 1 ) −1 = S. Now suppose that r > 1, then by the same argument as for the case r = 1 we obtain that d 1 ∈ S and thus c 1 [Proof of Claim A] Using Claim A, we prove Claim B: For all α ∈ S \ S × we have a primary decomposition of (X α ) with associated primes of height one.
Proof of Claim B. Let α ∈ S \ S × . Then X α is only contained in maximal t-ideals of the form K[P ] for P ∈ X(S), since if Q is a maximal t-ideal containing X α , then Q ∩ S is a non-empty prime t-ideal. One can see this as follows: Clearly, K[Q ∩ S] ⊆ Q and by [11,Lemma 2 , thus equality holds. Since S is weakly Krull, α is only contained in finitely many height-one primes of S, say P 1 , . . . , P n . It follows that the only maximal t-ideals of K[S] containing X α are the K[P 1 ], . . . , K[P n ]. Now f , hence without loss of generality we write f g = f X α for appropriate α ∈ S and obtain f g Q = X α f Q . We show f ∈ (X α ). By Claim B we pick a primary decomposition ( . We are now in the position to prove our main result, characterizing the weakly Krull monoid algebras. Next (borrowing heavily from Chang [7]) we give an example of a group algebra over a field that is not weakly Krull, showing that the assumption of the group satisfying the ACC on cyclic subgroups cannot be omitted. Note that for a field K and a group G we always have that K is weakly Krull G-UMT and that G is weakly Krull K-UMT.
Example 3.8. Let K be a field and let Q be the additive group of rational numbers. For each n ∈ N let G n be the subgroup of Q generated by 1 2 n .
Proof. 1. Since G n ∼ = Z as additive groups, we have K[G n ] ∼ = K[Z], which is well known to be a PID. 2. Follows from the fact that 1 + y is a prime element in K[y] and that prime elements are lifted to prime elements under localization, provided they do not become units. 3. If we set y = X 1 2 n and k = 2 n−m , then y is an indeterminate over K and K[G n ] ∼ = K[y, y −1 ]. Now 1 + y k / ∈ (1 + y)K[y] since k is even, thus 1 + y k / ∈ (1 + y)K[y, y −1 ], because (1 + y)K[y] is a prime ideal. 4. By [15,Corollary 12.11.1] it suffices to prove that G n ⊆ Q is a root extension, to prove that K[G n ] ⊆ K[Q] is an integral extension and hence K[Q] is of Krull dimension 1 by 1. and thus of t-dimension 1. To prove that we have a root extension, let a b ∈ Q with b ∈ N. Then b a b = a = (2 n a) 1 2 n ∈ G n . 5. We show that 1 − X is in infinitely many height one prime ideals of K[Q]. Clearly, 1 − X = (1 + X 1 2 )(1 + X 1 2 2 ) · · · (1 + X 1 2 n )(1 − X 1 2 n ), so it suffices to show that for m = n the elements 1 + X 1 2 n and 1 + X 1 2 m are never in the same prime ideal, since by t-dimension is 1 they are contained in height-one prime ideals. Assume to the contrary that this was the case, say P is a prime ideal containing 1 + X

Applications
In this section, we investigate the K-(resp. G-) UMT property for special monoids (resp. domains). Based on these results, we give applications of our main result Theorem 3.7. We start with non-negative monoids of totally ordered abelian groups, whose monoid algebras were recently studied in [10] and [18].
Lemma 4.1. Let (G, ≤) be a totally ordered abelian group satisfying the ACC on cyclic subgroups and S = {g ∈ G | g ≥ 0} be the non-negative monoid of G. Then S is K-UMT for all fields K.
Proof. Assume to the contrary, that S is not K-UMT for some field K. Then there is P ∈ X(S) such Clearly, A has no monomials, so has to be of the form ." Now let f ∈ K[G] be as above and let s 1 be the smallest exponent of f . Then let s ∈ G with s + s 1 = 0, thus f X s =: g ∈ K[S] and f | K[G] g but g / ∈ K[P ]; a contradiction. We are now able to prove that the classical UMT property is equivalent to our notion of being N 0 -UMT, provided the domain is weakly Krull. For this, we need the following preparatory Proof. Since D is weakly Krull and N 0 -UMT it suffices to show that N 0 is weakly Krull and K-UMT in order to apply Theorem 3.7, but that N 0 is weakly Krull is well known (it is factorial) and the property of being K-UMT was shown in Lemma 4.1. (a) D is UMT, Proof. Next we apply our main result to monoid algebras over affine monoids, which for example occur (over fields) when studying polytopes (see [6]). Recall that an affine monoid is a torsion-free finitely generated monoid.
Lemma 4.5. Let S be an affine monoid. Then there is n ∈ N such that the quotient group q(S) ∼ = Z n . In particular, q(S) satisfies the ACC on cyclic subgroups.
Proof. Since S is a finitely generated submonoid of Z m for some m ∈ N, q(S) is a Z-submodule of Z m and therefore is itself a free Z-module of rank n ≤ m.
The following remark is well known, but for the convenience of the reader and since the argument is short, we give it.
Remark 4.6. Let S be a finitely generated monoid, then its root closure S is Krull.
Proof. Note that if S is finitely generated, then also its reduced monoid S red is, so by [  Proof. Let S be an affine monoid with quotient group G and let K be a field. In Lemma 4.5 we just proved that G satisfies the ACC on cyclic subgroups, thus in particular S × does. Since S is finitely generated, S is Krull by the above remark. It follows by [ Proof. To begin with, note that by Lemma 4.5 G := q(S) ∼ = Z n satisfies the ACC on cyclic subgroups and that D being UMT is equivalent to D being G-UMT by Proposition 4.4. It remains to prove that S is D-UMT, which is equivalent to S being K-UMT by Lemma 3.4, where K is the quotient field of D, but this is just Lemma 4.7. Now apply Theorem 3.7.
Since fields are always UMT, an easy consequence of this proposition is the following We now deduce the results by Chouinard resp. El Baghdadi and Kim on Krull resp. generalized Krull monoid algebras from our main result Theorem 3.7. Next we prove that S is D-UMT. Since S is a Krull monoid, for every P ∈ X(S) there is a discrete rank-one valuation w P : G → Z with valuation monoid S P . Again, for P ∈ X(S) we extend w P to a discrete rank-one valuation on the quotient field of D is weakly Krull with trivial t-class group. It follows by Theorem 3.7 that D and S are weakly Krull and that they are GCD follows by a fact on splitting sets, see the proofs of [7,Lemmata 7 and 8]. It remains to prove that their t-class groups are trivial. But it is well-known that their t-class groups embed into the t-class group of D[S] (see for example [11]).