On the distribution of prime divisors in Krull monoid algebras

In the present work, we prove that every class of the divisor class group of a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in the literature so far, unfortunately with open gaps. We present a complete proof of this fact.


Introduction
The investigation of class groups and the distribution of prime divisors in the classes is a central topic in ring theory and has been studied for many classes of rings, e.g. for orders in number fields. A variety of realization theorems for class groups have been achieved, for instance that every abelian group occurs as the class group of a Dedekind domain (Claborn's Realization Theorem) respectively as the class group of a simple Dedekind domain [18]. In the present paper, we study the distribution of prime divisors in the class group of commutative monoid algebras D[S] that are Krull. In our setting, a monoid S is a cancellative commutative unitary semigroup. For a commutative ring D with unity and a monoid S, we denote by D[S] the monoid algebra of D over S.

It is a well-known result by Chouinard, that D[S] is a Krull domain if and only if D is a Krull domain
and S is a torsion-free Krull monoid with S × satisfying the ACC (ascending chain condition) on cyclic subgroups, where S × denotes the group of units of S (see [5]). Moreover, we have a canonical isomorphism of divisor class groups In the setting of monoids, it is known that every abelian group is the class group of a Krull monoid and the possible ways of distributing the prime divisors in the classes are completely determined [8,Theorem 2.5.4]. Considering domains, a slightly weaker form of a realization result holds true. Although every abelian group is the class group of a Dedekind domain, it is not completely clear how prime divisors behave [8,Theorem 3.7.8].
In Section 2, we fix notations concerning monoid algebras and class groups. Section 3 contains a proof of the following result concerning the distribution of prime divisors: Let K be the quotient field of D and let G be the quotient group of S.
with d i ∈ K and s i ∈ G we denote by A f resp. E f the fractional ideal of D resp. S generated by d 1 , . . . , d n resp. s 1 , . . . , s n . Let G be an abelian group. An element g ∈ G is said to be of height (0, 0, 0, . . .) if the equation px = g has no solution x ∈ G for any prime number p. Furthermore, g is said to be of type (0, 0, 0, . . .) if the same equation has a solution for only finitely many prime numbers. The group G is said to be of type (0, 0, 0, . . .) if every element of G is of type (0, 0, 0, . . .). It is well known that G is of type (0, 0, 0, . . .) if and only if it satisfies the ACC (ascending chain condition) on cyclic subgroups. For a proof of this statement and for further equivalent conditions, see [10, §14]. Let S be a monoid. By C t (S) we denote the the t-class group of S which is the quotient group of the group of t-invertible fractional t-ideals of S modulo the subgroup of non-empty principal fractional ideals of S. If S is a Krull monoid, then C t (S) coincides with the v-class group C v (S) (which is isomorphic to the divisor class group of S). Moreover, in this case C v (S) is isomorphic to the quotient of the monoid of v-invertible (integral) v-ideals modulo the submonoid of non-empty principal (integral) ideals. If I is a t-invertible fractional t-ideal of S, we denote by [I] ∈ C t (S) its t-class and refer to it as the divisor class of I if S is a Krull monoid. Moreover, we denote by X(S) the set of height-one prime ideals of S. In the literature, the elements of X(S) are also often called prime divisors of S (especially, if S is a Krull monoid). Replacing the monoid S by a domain D, the terminology and notation concerning class groups and prime divisors above is the same. For further information, see [8]. We will make intensive use of the theory of valuations on fields and groups. If D is a domain and P is a prime ideal of D such that the localization D P is a valuation domain, then we denote by v P its valuation induced on the quotient field of D and call it the P -adic valuation. In the case of monoids, the notation is analogous. The interested reader is referred to [2] and [10]. For the remainder of this work and if not specified otherwise, D denotes a domain with quotient field K and S is a torsion-free monoid with quotient group G.

Proof of the main result
To ensure the existence of prime divisor in divisor classes of D[S], it is necessary to construct irreducible elements in K[G]. The following lemma (which is due to Matsuda [16]) is a first step into this direction that will be used in a special case. For the convenience of the reader we include the full proof.
Let H be the subgroup of G that is generated by g together with the exponents of f and h. We first show, that the subgroup g of H generated by g is a direct summand of H. Let p be a prime number. Suppose p n x ∈ g for some positive integer n and x ∈ H. Then there exists m ∈ Z such that p n x = mg. Since H is torsion-free, there are integers n ′ ≥ 0 and m ′ such that p n ′ x = m ′ g and gcd(p, m ′ ) = 1. It follows that 1 = kp n ′ + lm ′ for some k, l ∈ Z. Multiplying by g gives g = kp n ′ g + lm ′ g = p n ′ (kg + lx). Since the height of g equals (0, 0, 0, . . .), we have n ′ = 0, hence x = m ′ g ∈ g . So we have shown that g is a pure subgroup of the finitely generated group H and is therefore a direct summand by [6,Corollary 25.3]. So we write H = g ⊕ e 1 ⊕ . . . ⊕ e n for some e 1 , . . . , e n ∈ H. The set {X g , X e1 , . . . , X en } is algebraically independent over K. Hence a + bX g is irreducible in the polynomial ring K[X g , X e1 , . . . , X en ] and therefore also in

It follows that either f or h is a unit in K[H] and hence in K[G].
The next result is a special case of [4,Theorem 3]. Nevertheless, we recall its proof.

Lemma 3.2. [4, Theorem 3] Let D[S] be a Krull monoid algebra and f ∈ K[G]\{0}. Then f K[G]∩D[S]
If I −1 is a principal fractional ideal, I is a principal ideal, hence we can assume without loss of generality that b is its generator. Now take any P ∈ X(D) and any element a ∈ D with v P (a) = 1. In particular, if D is not semi-local, there exist infinitely many such P . Now assume that I −1 is not a principal fractional ideal. Thus, there exists Moreover v P ( a b ) = 1. If D is not semi-local, then X(D) is infinite, whence there are infinitely many P ∈ X(D) such that v P (a ′ ) = 0 = v P (b). For each of them, we construct a in the following way using the Approximation Theorem for Krull domains: [13] showed that every divisor class of D[G] (where G = {0}) contains a prime divisor. We copy and modify his proof in such a way that the existence of infinitely many prime divisors in each class follows.

Lemma 3.4. Let G be a non-zero abelian group such that D[G] is a Krull domain. Then each divisor class of D[G] contains infinitely many prime divisors.
Proof. Note that G is torsion-free, because D[G] is Krull.   . Therefore, if S is non-reduced then S × is non-trivial and we can use Lemma 3.4 to show that every divisor class contains infinitely many prime divisors. So, from now on, assume that S is reduced. Let G be the quotient group of S. Case 1: First assume that S is finitely generated. Then, since S is torsion-free, it is isomorphic to an additive submonoid of the group (Z m , +) for some m ∈ N [3, page 50]. Therefore G is a Z-submodule of the free Z-module Z m and hence free, say of rank n ∈ N.
Proof of Claim A. If n = 0 or n = 1, this is trivial. So let n > 1. Then in particular S = G, because S is reduced. Let P ⊆ S be a height-one prime ideal of S and v P : G → Z the associated P -adic valuation. Let a ∈ S with v P (a) = 1. Let G 0 be the kernel of v P . Then G 0 is a free Z-module of rank n − 1 and we have G ∼ = ker v P ⊕ im v P ∼ = G 0 ⊕ Z. So, if a 1 , . . . , a n−1 is a Z-basis of G 0 , then a 1 , . . . , a n−1 , a is a Z-basis of G with a ∈ S.
[Proof of Claim A] Now let (a 1 , . . . , a n ) be a Z-basis of G with a 1 ∈ S. We define O = K[a 1 ] with quotient field F = K(a 1 ). By L = K(a 1 , . . . , a n ) we denote the quotient field of K . . , g n ∈ G with I −1 = ((g 1 + S) ∪ . . . ∪ (g n + S)) v . Since S is a reduced non-finitely generated Krull monoid, it follows from [8, Theorem 2.7.14] that the set X(S) of height-one prime ideals of S is infinite. Thus, there exist infinitely many P ∈ X(S) such that 0 = v P (g 1 ) = . . . = v P (g n ). For each choice of P , we construct a prime divisor lying in [K[I]] (and infinitely many of them are pairwise distinct). Let a ∈ S with v P (a) = 1. Let g = X g1 + . . . + X gn + X gn+a ∈ K[G].

Claim B: g is irreducible in K[G].
Suppose that the claim holds true. On the one hand, I −1 = ((g 1 + S) ∪ . . . ∪ (g n + S)) v = ((g 1 + S) ∪ . . . ∪ (g n + S) ∪ (g n + a + S)) v , because a ∈ S. On the other hand, by Lemma 3. Proof of Claim B. It suffices to show that g is irreducible in K[S P ], because K[G] is the localization of K[S P ] at the set {X s | s ∈ P } and K[S P ] is a factorial domain. To see that g ∈ K[S P ] is irreducible, let h 1 , h 2 ∈ K[S P ] with g = h 1 h 2 . Since S P is a discrete rank one valuation monoid, the map N 0 × S × P → S P via (n, s) → sa n is an isomorphism (note that a ∈ S P was chosen with v P (a) = 1). We endow N 0 × S × P with a total order compatible with the monoid operation in the following way: N 0 carries the canonical order ≤. On S × P take any total order ≤ compatible with the group operation, which is possible by [10,Corollary 3.4]. Now N 0 × S × P is a totally ordered monoid with lexicographic order, i.e., for m, n ∈ N 0 and s, t ∈ S × P . Now we can write . Also g = X (0,g1) + . . . + X (0,gn) + X (1,gn) in this notation. It follows that (1, g n ) = (a n , b n ) + (c m , d m ) = (a n + c m , b n + d m ), thus 1 = a n + c m . Without loss of generality, we can suppose that a n = 0 and c m = 1. Therefore a i = 0 for all i ∈ {1, . . . , n}. Let j ∈ {1, . . . , m} be minimal such that c j = 1. Then the monomial in h 1 h 2 = g with exponent (a 1 , b 1 ) + (1, d j ) = (1, b 1 + d j ) has a non-zero coefficient and therefore it holds that (a 1 , b 1 . It follows that n = 1 and hence h 1 = k 1 X (0,b1) ∈ K[S P ] is a unit. This proves that g ∈ K[S P ] is irreducible. [

Past proof attempts
As noted in the beginning of this work, Chang [4] states the fact that in a Krull monoid algebra D[S] every divisor class contains a prime divisor. Unfortunately, due to an error in Lemma 7 of his work (Krull monoids in general do not satisfy the approximation property, see [11,Theorem 26.4]), the proof of his main result collapses. Another error that occurs in Chang's argumentation is due to a mistake in [13,Corollary 5], where Kim asserts that fractional v-ideals of Krull monoids S such that D[S] is a Krull domain are always v-generated by two elements. A counterexample to this assertion can be found in [17,Beispiel 3.8]. In the following, we give an explicit counterexample to Chang's Lemma 7.