Noetherian semigroup algebras and prime maximal orders

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Noetherian semigroup algebras and prime maximal orders

Isabel Goffa

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Abstract

Let S be a semigroup and K be a field. A K-space K[S], with basis S and with multiplication extending, in a natural way, the operation on S, is called a semigroup algebra. It remains an open problem to characterize semigroup algebras that are a prime Noetherian maximal order.

In this thesis, we give an answer to the problem for a large class of cancellative semigroups and we illustrate these results with several examples of concrete classes of Noetherian maximal orders. Indeed, we find necessary and sufficient conditions for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group to be a maximal order. Un- der an invariance condition on the minimal primes, our result is entirely in terms of the monoid S and, in order to prove it, we describe the height one prime ideals of K[S]. Recall that it is conjectured that polycyclic-by- finite groups G are the only groups having a Noetherian group algebra and K.A. Brown characterized when these group algebras K[G] are a prime Noetherian maximal order.

In case K[S] also satisfies a polynomial identity, this means in case S is a submonoid of a finitely generated abelian-by-finite group, we show that the invariance condition on the minimal primes of S is necessary for K[S] to be a prime Noetherian maximal order. Furthermore, we establish a general method for constructing non-abelian maximal order semigroup algebras of finitely generated submonoids of abelian-by-finite groups, starting from abelian maximal orders. To obtain concrete constructions, we thus also need to deal with abelian finitely presented monoids A. If A has a presentation with one or two defining relations, we determine necessary and sufficient conditions for K[A] to be a domain that is a maximal order.

The description is only in terms of the defining relations. Furthermore, we compute the class groups of such semigroup algebras.

In the appendix, we briefly explain applications of maximal orders in space-time coding. These applications softly point out that maximal orders not only might be interesting for experts in algebra, but also for specialists in coding theory.

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Dankwoord

Mijn grootste dank ben ik verschuldigd aan mijn promotor Prof. Dr. Eric Jespers. Ontzettend dankbaar ben ik hem omdat hij mij de kans gaf dit doctoraat te schrijven. Het is bewonderenswaardig hoe ver en diep zijn kennis over dit onderzoek reikt en hoe geduldig, goed en vriendelijk hij die duidelijke kijk met mij wist te delen. De vele congressen en studiever- blijven die ik met hem heb mogen meemaken waren telkens zeer leerrijk en bijzonder aangenaam. Dankzij hem kreeg ik de kans talrijke interessante en sympathieke wiskundigen te ontmoeten en de mooie herinnering aan menig amusante congresavond zal mij altijd bijblijven.

I would like to warmly thank Prof. Dr. Jan Okni´nski. Thank you for the cooperation and for the willingness to extensively answer to all my questions. I am grateful for the papers we wrote together with Eric and for the interesting conversations we had.

Bedankt aan het Instituut voor de Aanmoediging van Innovatie door Wetenschap en Technologie in Vlaanderen, dat mij het mogelijk maakte om vier jaar lang onderzoek te kunnen verrichten aan de Vrije Universiteit Brussel. Telkens was alles tot in de puntjes geregeld en ik ben blij dat ik via hen de uiterst vriendelijke Prof. Dr. Alfons Ooms heb leren kennen.

Een thesis schrijf je niet alleen. Op mijn weg ben ik tal van mensen tegengekomen, die door hun blijk van interesse, er telkens opnieuw in slaag- den mij aan te sporen. Zo ben ik de collega’s van het wiskunde departement een speciaal woordje van dank verschuldigd. Een koffiepauze zo nu en dan, een gezellige lunch, het zijn de dingen die elke werkdag telkens opnieuw aangenaam maken. De klasgenootjes tijdens mijn licentiaatsopleiding, die later mijn collega’s werden, zou ik willen danken voor alle ontspannende momenten en voor de vele motiverende babbels die we samen hadden. Dank ook aan Philippe, die interessante seminaries gaf over space-time coding.

Mijn familie en vrienden wil ik uitermate danken omdat ze er steeds voor me zijn en omdat ik altijd op hen kan rekenen. Joris, omdat je bent wie je bent en omdat je mij steunt in alles wat ik doe, ben ik jou ontzettend

dankbaar. Isabel Goffa

Februari 2008 iii

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Do not spoil what you have by desiring what you have not; but remember that what you now have was once among the things you only hoped for.

Epicurus

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Chapter 1 Introduction

The aim of the work in this thesis is to present new classes of algebras with a rich algebraic structure, such as prime Noetherian maximal orders. Because of the role of Noetherian rings in the algebraic approach in noncommutative geometry, new concrete classes of finitely presented algebras recently gained a lot of interest. Via the applications to the solutions of the Yang-Baxter equation, one such class also widens the interest into other fields, such as mathematical physics, and therefore these algebras have been extensively studied, see for example [20, 21, 25, 27, 39, 68].

In this thesis, constructions are given of finitely presented algebras over a field K defined by monomial relations. In order to describe when these algebras are Noetherian maximal orders, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra.

Hence, it is natural to consider such algebras as semigroup algebras K[S]

and to investigate the structure of the monoid S. Furthermore, it turns out that many of these algebras are subalgebras of group algebras of polycyclic- by-finite groups. This link to group algebras, that is of interest, yields a rich resource of constructions not only for the non-commutative ring theorists, but also to researchers in semigroup theory and certain aspects of group theory.

In general, it remain unsolved problems to characterize when an arbitrary semigroup algebra K[S] is Noetherian and when it is a prime Noethe- rian maximal order. Recall that the former question even for group algebras has been unresolved. The only class of groups for which a positive answer has been given is that of the polycyclic-by-finite groups, i.e. it is conjectured that these groups are the only groups that yield a Noetherian group algebra.

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Brown ([6, 7]) characterized when such group algebras are prime Noethe- rian maximal orders and when they are unique factorization rings or domains, in the sense as studied by Chatters and Jordan ([12]).

Hence, in the context of semigroup algebras, it is natural to consider submonoids of polycyclic-by-finite groups. Jespers and Okni´nski [45] characterized when the semigroup algebra of a submonoid of a polycyclic-by-finite group is right Noetherian (Theorem 3.1.4). It turns out that these semigroup algebras are right Noetherian if and only if they are left Noetherian and that, if K[S] is right Noetherian, the semigroup S is finitely generated.

For general semigroup algebras it is not known whether K[S] right Noethe- rian implies S finitely generated. In [60], the author gives some classes of semigroups that satisfy this implication. Furthermore, if S is an abelian monoid, Gilmer [28] proved that the semigroup algebra K[S] is Noetherian if and only if the semigroup S is finitely generated.

Concerning the maximal order problem, if S is an abelian monoid, An- derson [1, 2] proved that K[S] is a prime Noetherian maximal order if and only if S is a submonoid of a finitely generated torsion-free abelian group and S is a maximal order in its group of quotients. More generally, Chouinard [14] proved that a commutative monoid algebra K[S] is a Krull domain if and only if S is a submonoid of a torsion-free abelian group which satisfies the ascending chain condition on cyclic subgroups and S is a Krull order in its group of quotients. In particular, it turns out that the height one prime ideals of K[S] determined by the minimal primes of S are crucial and that the class group of K[S] is isomorphic to the class group of S.

Apart from the cases mentioned above, an answer to the question has been obtained only for some special classes of cancellative semigroups, such as submonoids of torsion-free finitely generated abelian-by-finite groups ([41]), or for some special classes of maximal orders, such as principal ideal rings ([38, 49]) and unique factorization rings ([47, 48]).

In this thesis we continue these investigations, we investigate when a semigroup algebra of a submonoid of a polycyclic-by-finite group is a prime Noetherian maximal order and we describe height one prime ideals of such semigroup algebras. So, for a large class of cancellative semigroups, we give an answer to the open question posed above.

We now briefly outline the content of each chapter. Chapter 2 is in- cluded for the convenience of the reader. This chapter contains the main prerequisites of semigroup and ring theory used in the thesis.

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3

In Chapter 3 we elaborate on our joint work with Jespers and Okni´nski and we characterize, under an invariance condition on the minimal primes, Noetherian semigroup algebras that are prime maximal orders, for submonoids of polycyclic-by-finite groups (Theorem 3.4.3). Knowledge of prime ideals turns out to be crucial and therefore we first describe the height one prime ideals of such prime Noetherian semigroup algebras (Theorem 3.2.2).

If K[S] is a domain, i.e. if SS⁻¹ is torsion-free, the same information on height one primes, and even information on primes that are not of height one, was obtained earlier by Jespers and Okni´nski in [42, 45]. However, we show in this thesis that the information on primes that are not of height one does not remain valid in case K[S] is prime and not a domain (Exam- ple 4.3.2).

As a consequence of the description of the height one prime ideals, we establish going up and going down properties between prime ideals of S and prime ideals of S ∩ H, where H is a subgroup of finite index in G = SS⁻¹ (Corollary 3.2.5). These are the analogs of the important results known on the prime ideal behaviour between a ring graded by a finite group and its homogeneous component of degree e (the identity of the grading group). As an application, it is shown that the classical Krull dimension clKdim(K[S]) of K[S] is the sum of the prime dimension of S and the plinth length of the unit group U(S) (Corollary 3.3.1). Also, a result of Schelter is extended to the monoid S: the prime dimension of S is the sum of the height and depth of any prime ideal of S (Proposition 3.3.3).

Furthermore, as a consequence of the characterization obtained in Chap- ter 3, we describe in Chapter 4 when a semigroup algebra of a submonoid of a polycyclic-by-finite group is a prime Noetherian maximal order that satisfies a polynomial identity, this means we characterize when a semigroup algebra of a submonoid of a finitely generated abelian-by-finite group is a prime Noetherian maximal order (Theorem 4.1.3). This characterization extends the earlier result, Theorem 3.1.6, in case the group of quotients is torsion-free. Furthermore, if the polycyclic-by-finite group in Theorem 3.4.3 is abelian-by-finite, we show that the invariance condition on the minimal primes in Theorem 3.4.3 is not only sufficient, but is also necessary. In Sec- tion 4.2, we illustrate Theorem 4.1.3 with an example of a prime Noetherian maximal order semigroup algebra (Example 4.3.2). As said above, this example also shows that Theorem 3.2.2 cannot be extended to prime ideals of height exceeding one. The obtained results in Chapters 3 and 4 can be found in the papers [30] and [31].

Introduction

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Theorem 4.1.3 reduces the problem of determining when K[S] is a prime Noetherian maximal order to the algebraic structure of S. It hence provides a strong tool for constructing new classes of such algebras. For some examples the required conditions on S can easily be verified, but on the other hand, for some other examples this still requires substantial work. That is why we establish a useful general method for constructing nonabelian submonoids of abelian-by-finite groups that are maximal orders, starting from abelian maximal orders (Proposition 4.2.1).

Anderson, Gilmer and Chouinard characterized abelian finitely generated monoids that are maximal orders ([1, 2, 14, 28]). Despite this nice and useful structural characterization, the following remains a challenging problem for a finitely generated submonoid S of a torsion-free abelian group:

determine necessary and sufficient conditions on the defining relations for S to be a maximal order. In Chapter 5, we precisely solve this problem in case S is defined by one or two relations (Theorem 5.1.2 and Theorem 5.2.2).

So, in contrast to the structural description obtained by Anderson, Gilmer and Chouinard, our contribution is a computational approach (based on presentations) to obtain a description of maximal orders defined via monomial relations. The advantage is that it also allows us to compute the class group cl(S) of such monoids S and thus of their algebras K[S] (Theo- rem 5.1.4 and Theorem 5.2.4). This group is the basic tool in the study of arithmetics of maximal orders ([23]) and hence it follows that our obtained results on abelian maximal orders and class groups, which are written in the paper [32], might be interesting for experts in related areas.

In Chapter 6 we illustrate the theory developed in the preceding chapters on several concrete examples of semigroup algebras and we come back on the non-commutative algebras, mentioned in the beginning of this introduction, that share many properties with polynomial algebras in finitely many commuting variables. Namely, we consider monoids of I-type, in- troduced by Gateva-Ivanova and Van den Bergh in [27], and later also studied by Jespers and Okni´nski in [43]. We reprove, with the help of Theorem 4.1.3, that their semigroup algebras are Noetherian maximal orders and domains.

Furthermore, we generalize the concept of a monoid of I-type to that of a monoid of IG-type and we characterize, again with the help of Theo- rem 4.1.3, when such semigroup algebras are prime Noetherian maximal orders. All the results on monoids of IG-type are published in our joint paper [29] with Jespers. A few other examples of maximal orders are de- scribed in Chapter 6.

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Finally, in the Appendix, we explain applications of maximal orders in space-time coding. We briefly outline the quite recently developed theory of space-time codes and we explain how maximal orders can be used to obtain new results in the field. These applications softly point out that maximal orders not only might be interesting for experts in algebra, but also for specialists in coding theory.

Introduction

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Chapter 2 Prerequisites on semigroup and ring theory

This chapter is devoted to the necessary background on semigroup, group and ring theory. The reader who is familiar with all the required concepts can omit this chapter. The material is a mixture of classical results that can be found in [45, 58, 60, 64, 65].

2.1 Semigroups and semigroup algebras

This section is an attempt to bring together the basic definitions and major results we will need from the theory of semigroups and semigroup algebras.

Some notions in semigroup theory can be derived from group theory, but most of them resemble ring theory more closely. Some notions are based on familiar concepts in group and ring theory. Our notation and terminology follows closely that of [15], [45] and [60].

A semigroup S is a non-empty set with an associative binary operation.

We will denote this operation multiplicatively. If S has an identity 1 then it is called a monoid and we write U(S) for the group {s ∈ S | sr = rs = 1 for some r ∈ S}, which is called the unit group of S. A monoid with S = U(S) is simply a group. An element θ of a semigroup S is a zero of S if it is a left and right zero, that is θs = sθ = θ for all s ∈ S. A null semigroup is a semigroup with zero in which the product of any two elements is the zero element θ.

If S is a semigroup without an identity, one can adjoin an identity by simply adding a new element 1 and extending the multiplication by defining

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2.1. Semigroups and semigroup algebras 8

1s = s1 = s for all s ∈ S ∪ {1}. We denote this new semigroup by S¹. If S already has an identity, then one agrees that S¹ = S.

In a very similar way, one can always adjoin a zero element θ, and we write

S⁰ = S ∪ {θ}

for this new semigroup.

A subsemigroup of a semigroup S is a non-empty subset which is closed under multiplication. A submonoid is a subsemigroup which has an identity.

A subgroup G of S is a subsemigroup which is a group. Note that even if S is a monoid, then G does not necessarily have the same identity element as S.

For a semigroup S, the subset

Z(S) = {x | xs = sx for all s ∈ S}

is called the centre of S and it is a commutative subsemigroup of S, that is, any two elements commute.

We now recall the definition of a semigroup ring.

Definition 2.1.1. Let R be a ring and let S be a semigroup. The semigroup ring R[S] is the ring whose elements are all formal sums P

s∈Sr_ss, with each coefficient rs ∈ R and all but finitely many of the coefficients equal zero. Addition is defined component-wise so that

X

s∈S

r_ss +X

s∈S

q_ss =X

s∈S

(r_s+ q_s)s.

Multiplication is given by

X

s∈S

rss

! X

s∈S

qss

!

=X

s∈S



 X

uv=s; u,v∈S

ruqv



s.

If R = K is a field, then K[S] is called a semigroup algebra.

For an element α = P

s∈Srss ∈ R[S], with each rs ∈ R, we put supp(α) = {s ∈ S | r_s6= 0}, the support of α.

If S is a semigroup without a zero θ it is clear that R[S⁰] = R[T ] × Rθ ∼= R[S] × Rθ,

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a direct product of rings, where T = {s − θ | s ∈ S} is a semigroup isomorphic with S. In the case S has a zero element θ, we write R₀[S] for the quotient ring

R[S]/Rθ.

The ring R0[S] is called the contracted semigroup ring of S over R. If R = K is a field then K₀[S] is called the contracted semigroup algebra.

Note that K0[S ∪ {θ}] ∼= K[S]. Hence every semigroup algebra can be seen as a contracted semigroup algebra.

Definition 2.1.2. If T is a non-empty subset of a monoid S, we write hT i for the submonoid generated by T and gr(T ) for the subgroup generated by T . If T is finite, say T = {t₁, . . . , t_n}, we often write ht₁, . . . , t_ni or gr(t1, . . . , tn). A monoid S is cyclic if S = hxi for some x ∈ S. An element x of a monoid S is a periodic element if hxi is finite.

Obviously, a semigroup homomorphism from a semigroup S to a semigroup T is a function f : S −→ T such that f (xy) = f (x)f (y) for all x, y ∈ S.

Let I be a non-empty subset of a semigroup S. Then I is a right ideal of S if xs ∈ I for all s ∈ S and x ∈ I. A left ideal is defined similarly and I is an ideal of S if it is a left and right ideal of S. If a ∈ S, then the ideal generated by a, that is, the smallest ideal of S containing a, is

J_a= S¹aS¹ = SaS ∪ Sa ∪ aS ∪ {a}.

We say that a semigroup S is simple if it has no ideals other than S itself. This definition is not interesting in case S has a zero element θ, since {θ} always is an ideal of S. Therefore we say that a semigroup with a zero θ is 0-simple if S² = S and S has no ideals other than S and {θ}.

Recall that a semigroup with zero θ is a nil semigroup if every s ∈ S is nilpotent, so there exists k ∈ N0 with s^k= θ. If S^k= {θ} for some k ∈ N0, then S is called nilpotent. A (left) ideal L of a semigroup with zero θ is nil if all its elements are nilpotent. L is said to be nilpotent if L^k = θ for some k ∈ N0 (that is, l1. . . lk = θ for all li ∈ L). The following theorem is taken from [22, Theorem 17.22]. It provides the analog for semigroups of a Levitzki-theorem.

Theorem 2.1.3. Let S be a monoid with a zero element. If S satisfies the ascending chain condition on one-sided ideals then every nil ideal of S is nilpotent.

Prerequisites on semigroup and ring theory

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For a relation ρ on a set S and s, t ∈ S we write (s, t) ∈ ρ also as s ρ t.

An equivalence relation ρ on a semigroup S is called a congruence if for any s, t, x in S we have sx ρ tx and xs ρ xt whenever s ρ t. We write S/ρ for the set of ρ-classes in S. It also is a semigroup for the natural operation induced by that of S.

Definition 2.1.4. Let S be a semigroup, ρ a congruence relation and S/ρ the semigroup of ρ-classes in S. Denote by φ_ρ : S −→ S/ρ the natural semigroup homomorphism. Its K-linear extension to an algebra epimor- phism K[S] −→ K[S/ρ] we also denote by φ_ρ and its kernel is the ideal I(ρ) of K[S] generated by the set {s − t | s, t ∈ S with s ρ t}. Hence

K[S/ρ] ∼= K[S]/I(ρ) as K-algebras.

We give three examples of congruence relations that are relevant in our work. First, let S be a submonoid of a group G. Assume F is a normal subgroup of G so that F ⊆ S. Denote by ∼_F the relation on S defined as follows:

s ∼F t if s = f t for some f ∈ F.

Because F is normal in G it is easily seen that ∼_F is a congruence relation on S.

Second, let I be an ideal in an arbitrary semigroup S and let ρ denote the congruence on S defined by s ρ t if s = t or s, t ∈ I. The semigroup S/ρ is usually denoted by S/I and is called the Rees factor of S by I. Also, for a field K, it is easily verified (see [60, Lemma 4.7]) that

K[S]/K[I] ∼= K0[S/I].

Intuitively, when we pass from S to S/I, we have identified all the elements of I with zero. As a special case, it is convenient to agree that S/∅ = S, even though ∅ is not an ideal. There is a one-to-one correspondence between the ideals of S containing I and the ideals of S/I.

Lemma 2.1.5. If I ⊆ J are ideals of S, then J/I is an ideal of S/I and (S/I)/(J/I) ∼= S/J .

If T is a subsemigroup of S and I is an ideal of S which intersects T nontrivially, then T ∩ I is an ideal of T and T /(T ∩ I) ∼= (T ∪ I)/I.

Thirdly, we associate with an ideal J of K[S] a congruence relation ρ_J on S. So this is the opposite of the process explained in Definition 2.1.4.

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Definition 2.1.6. Let S be a semigroup and let J be an ideal of K[S].

Then, ρ_J denotes the congruence on S defined by ρ_J = {(s, t) | s, t ∈ S, s − t ∈ J }.

Note that I(ρJ) ⊆ J . There are natural epimorphisms K[S] −→ K[S/ρJ]

and

K[S/ρ_J] −→ K[S]/J.

One often identifies the semigroup S/ρ_J with its image in K[S]/J .

Throughout this thesis we are mainly interested in submonoids S of groups. Often the elements of these groups are fractions of elements of S.

In order to clearly describe this we need more terminology.

Definition 2.1.7. A semigroup S is called left cancellative (respectively right cancellative), if, for every a, b, x in S, xa = xb implies a = b (respectively ax = bx implies a = b). A semigroup is cancellative if it is left and right cancellative.

We say that S satisfies the right Ore condition if for every a, b ∈ S we have aS ∩ bS 6= ∅. It is well known that, for cancellative semigroups S, this happens if and only if S has a group of right quotients G = SS⁻¹ = {ab⁻¹| a, b ∈ S}, which is unique up to isomorphism. If S also has a group of left quotients H (defined symmetrically), then G, H are isomorphic and are referred to as the group of (two-sided) quotients of S.

In the next lemma we determine a sufficient condition for a cancellative semigroup to have a group of (two-sided) quotients ([45, Lemma 2.1.3]).

Lemma 2.1.8. Assume that S is a left cancellative semigroup with no free nonabelian subsemigroups. Then, for every right ideals I, J of S we have I ∩ J 6= ∅. In particular, if S is cancellative with no free nonabelian subsemigroups, then S has a group of (two-sided) quotients.

From Lemma 2.1.8 it follows that a submonoid S of an abelian-by-finite group G has a group of quotients. Recall that a group is abelian-by-finite if it has a normal abelian subgroup of finite index and thus, indeed, such a monoid can not have free nonabelian subsemigroups. In general, if P and Q are two group properties, a group G is said to be P -by-Q, if there exists a normal subgroup N of G that has property P and so that the quotient group G/N has property Q.

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2.2. Prime ideals in Noetherian and graded rings 12

Finally, we define the dimension of a semigroup, but before we do this, we give some more definitions. A semigroup S (with a zero element) is prime if IJ = {θ} for some ideals I, J of S implies that I = {θ} or J = {θ}. A proper ideal Q of a semigroup S (with a zero element) is a prime ideal if S/Q is a prime semigroup. Note that, by definition, prime ideals are different from S. A minimal prime ideal in S is a prime ideal of S that does not properly contain another prime ideal.

The prime spectrum Spec(S) is the set of all prime ideals of S. The rank rk(S) of a monoid S (not necessarily cancellative) is the supremum of the ranks of the free abelian subsemigroups of S (see for example [60]). So, for a finitely generated torsion-free abelian group A this corresponds with the classical notion of torsion-free rank of A. If S does not have a zero element then the height ht(Q) of a prime ideal Q is the maximal length n of a chain Q₀ ⊂ Q₁⊂ Q₂⊂ · · · ⊂ Q_n= Q, where Q₀= ∅ and Q₁, . . . , Q_n−1 are prime ideals of S. So, in this case, ht(Q) = 1 if and only if Q is a minimal prime ideal of S. On the other hand, if S has a zero element, then ht(Q) is the maximal length of such a chain with all Q_i prime ideals of S, i ≥ 0. If such n does not exist then we say that the height of Q is infinite. By X¹(S) we denote the set of all minimal prime ideals of S.

Definition 2.1.9. The dimension dim(S) of S is defined in the following way. By definition dim S = 0 if S = {e}. If S has no zero element, then dim(S) is the maximal length n of a chain Q₀ ⊂ Q₁ ⊂ Q₂ ⊂ · · · ⊂ Q_n, where Q₀ = ∅ and Q_i are prime ideals of S for i > 0, or ∞ if such n does not exist. If {e} 6= S has a zero element, then dim(S) is the maximal length n of such a chain with all Q_i (i ≥ 0) prime ideals of S, or ∞ if such n does not exist.

2.2 Prime ideals in Noetherian and graded rings

In this section we recall some basic terminology and some well known fundamental structural results on prime ideals in Noetherian and graded rings.

Several ring dimensions, such as classical Krull dimension and Gelfand- Kirillov dimension will be defined. For a survey and more detail we refer the reader to [45], [58] and [65].

A ring R will mean a nonzero associative ring, usually assumed to have an identity 1, except specified otherwise. Ring homomorphisms will be expected to preserve the identity and subrings to have the same identity.

A ring R is said to be right Noetherian (respectively right Artinian) if it satisfies the ascending (respectively descending) chain condition on right

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ideals. Similarly one defines left Noetherian and left Artinian rings. A Noetherian (respectively Artinian) ring is one that is both right and left Noetherian (respectively Artinian). A right Artinian ring is right Noethe- rian.

Recall that a ring R is said to be prime if the product of two nonzero ideals is nonzero. Obviously, in this case, the centre Z(R) of R is a domain, i.e. it has no zero-divisors. A proper ideal P of R is prime if R/P is a prime ring. A minimal prime ideal in R is a prime ideal of R that does not properly contain another prime ideal. Any prime ideal is contained in a maximal ideal and maximal ideals are prime ideals. A ring R is said to be simple if R has precisely two ideals, that is, {0} is a maximal ideal. A ring R is (right) semisimple if it is semisimple as a right R-module. A nonzero module M is said to be simple, if {0} and M are its only submodules. A module that is the direct sum of simple submodules is called semisimple.

A ring R is said to be semi-local if R/J (R) is semisimple, where J (R) is the Jacobson radical of R, i.e. the intersection of all maximal left ideals of R. Every nil ideal of R is contained in J (R). A (left) ideal L of a ring R is nil if all its elements are nilpotent. L is said to be nilpotent if Lⁿ = 0 for some n ∈ N0 (that is, l1. . . ln= 0 for all li ∈ L). The prime radical (or Baer radical) B(R) of a ring R is the intersection of the prime ideals of R.

A ring R is semiprime if and only if B(R) = {0}, and a proper ideal P of R is semiprime if R/P is a semiprime ring.

Theorem 2.2.1. (Levitzki) If R is a left Noetherian ring, then every nil left or right ideal is nilpotent.

The prime spectrum Spec(R) of a ring R is the set of all prime ideals of R. The height ht(P ) of a prime ideal P of R is the largest length of a chain of prime ideals contained in P . So ht(P ) = 0 means P is a minimal prime ideal. By X¹(R) we denote the set of all height one prime ideals of R.

The following theorem, which is called the principal ideal theorem ([58, Theorem 4.1.11, Corollary 4.1.12]), concerns height of primes in right Noethe- rian rings. This theorem will often be used in our further work. Recall that an element n of a ring R is a normal element if Rn = nR.

Theorem 2.2.2. Let R be a right Noetherian ring and n a normal element of R which is not a unit. If P is a prime ideal of R minimal over Rn then ht(P ) ≤ 1. More general, if Q is any prime ideal containing n then ht(Q) ≤ ht(Q/Rn) + 1.

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The classical Krull dimension clKdim(R) of R is the supremum of the lengths of all finite chains of prime ideals or ∞ if such chain does not exist.

We next recall the Gelfand-Kirillov dimension of an algebra. It measures the rate of growth of K-algebras in terms of any generating set and for a semigroup algebra K[S] it measures the rate of growth of the semigroup S.

Definition 2.2.3. Suppose R is as a K-algebra generated by a finite dimensional subspace V . Put R₀ = V⁰ = K and R_n = Pn

i=0Vⁱ for each positive integer n. Clearly each Rn is a finite dimensional K-space. The number

limsup_n→∞ log dim_KR_n log n

,

is independent of the choice of V . It is called the Gelfand-Kirillov dimension of R and it is denoted by GK(R). Now GK(R) = 0 means that R is a finite dimensional K-algebra. Furthermore, GK(R) is finite if and only if there exists a positive integer m so that dim_KR_n ≤ n^m for all sufficiently large positive integers n. For a not necessarily finitely generated K-algebra R one defines the Gelfand-Kirillov dimension as the supremum of all GK(R⁰), where R⁰ runs through all finitely generated K-subalgebras of R.

If K is a field and R a finitely generated commutative K-algebra, then GK(R) = clKdim(R).

In case a semigroup algebra K[S] of a cancellative semigroup S satisfies a polynomial identity, there is not only a nice link between the classical Krull dimension and the Gelfand-Kirillov dimension of K[S], but also between these dimensions and the rank of S (see for example [60, Theo- rem 23.4]). Recall that an algebra R over a field K satisfies a polynomial identity (abbreviated, R is a PI algebra) if there exists a nonzero polynomial f (x1, . . . , xn) in the free K-algebra on generators x1, . . . , xn so that f (r₁, . . . , r_n) = 0 for all r₁, . . . , r_n∈ R.

Theorem 2.2.4. Let S be a cancellative semigroup and K a field. If K[S]

satisfies a polynomial identity then clKdim(K[S]) = rk(S) = GK(K[S]).

We now recall a characterization of semigroup algebras of cancellative semigroups satisfying a polynomial identity (see [45, Theorem 3.1.9]).

Theorem 2.2.5. Let K be a field. If S is a cancellative semigroup then K[S] satisfies a polynomial identity if and only if S has a group of quotients G so that K[G] satisfies a polynomial identity. The latter holds if and only if G is abelian-by-finite and char(K) = 0 or G is (finite p-group)-by-(abelian- by- finite) and char(K) = p > 0.

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Another important theorem concerning height of primes, is Schelter’s theorem (see for example [58, Theorem 13.10.12]). An affine K-algebra is an algebra that is finitely generated as a K-algebra by a finite set of elements or, equivalently, by a finite dimensional K-subspace V .

Theorem 2.2.6. (Schelter) Let R be a prime PI affine K-algebra and P ∈ Spec(R). Then clKdim(R) = ht(P ) + clKdim(R/P ).

From Schelter’s theorem the following corollary easily follows. Recall that a ring R has the catenary property if given any P, P⁰ ∈ Spec(R) with P⁰ ⊆ P , any two saturated chains of primes between P and P⁰ have the same length. A chain is saturated if no additional term can be inserted.

Corollary 2.2.7. Each PI affine K-algebra has the catenary property.

To end this section, let us concentrate on prime ideals in graded rings.

Since several of the proofs on semigroup algebras given in this thesis even- tually rely on reductions to some graded rings, we need to recall some structural results in this general framework. There is a strong link between the primes of a finite group graded ring and those of its identity component (see [65]). However, we only state here the properties relevant for later use.

Let G be a group. A ring R is said to be G-graded if R = ⊕g∈GRg, the direct sum of additive subgroup Rg of R such that RgR_h ⊆ R_gh, for all g, h ∈ G. If always R_gR_h = R_gh then R is said to be strongly graded.

The additive groups Rg are called the homogeneous components of R and Re (also often denoted as R1) is the identity component, where e is the identity of the group G. The elements of ∪_g∈GR_g are called homogeneous.

Obviously a group algebra K[G] is an example of a strongly G-graded ring.

This ring, however, has many other natural gradings. For example if N is a normal subgroup of G and T is a transversal of N in G then K[G] is strongly G/N -graded with homogeneous components K[N ]t where t ∈ T . Actually, K[G] = K[N ] ∗ (G/N ), a crossed product of G/N over K[N ] (see [65] for the definition).

The following theorem is taken from [65, Theorem 17.9].

Theorem 2.2.8. Let R = ⊕g∈GRg be a ring graded by a finite group G with identity e.

1. Cutting Down If P ∈ Spec(R) then

P ∩ Re= Q1∩ · · · ∩ Q_n,

where n ≤ |G| and Q1, · · · , Qnare all the prime ideals of Re minimal over P ∩ Re. Furthermore, each prime Qi has the same height as P and P is maximal among all ideals I of R such that I ∩ R_e⊆ Q_i.

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2. Lying Over If Q ∈ Spec(Re) then there exists P ∈ Spec(R) (of the same height as Q) so that

P ∩ Re= Q1∩ · · · ∩ Q_n,

for some primes Q = Q1, · · · , Qn of Re minimal over P ∩ Re (again all of the same height as Q), and there are at most |G| such prime ideals. One says that P lies over Q.

3. Incomparability Given the lying over diagram, P₁

. ...

Q1 P2

... . Q2

where P₁ lies over Q₁, P₂ lies over Q₂ and P₂ ⊆ P₁, Q₂ ⊆ Q₁. If P₁ 6= P₂, then Q₁ 6= Q₂.

4. Going up Suppose Q₂ is a prime ideal of R_e and P₂ is a prime ideal of R lying over Q₂.

(a) If Q₁ is a prime ideal of R_e containing Q₂ then there exists a prime ideal P1 of R lying over Q1 so that P2 ⊆ P₁.

(b) If P1 is a prime ideal of R containing P2 then there exists a prime ideal Q₁ of R_e containing Q₂ so that P₁ lies over Q₁. 5. Going down Suppose Q1 is a prime ideal of Re and P1 is a prime

ideal of R lying over Q₁.

(a) If Q₂ is a prime ideal of R_e contained in Q₁ then there exists a prime ideal P₂ of R lying over Q₂ so that P₂ ⊆ P₁.

(b) If P2 is a prime ideal of R contained in P1 then there exists a prime ideal Q2 of Re contained in Q1 so that P2 lies over Q2. Furthermore clKdim(R) = clKdim(Re).

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2.3. Polycyclic-by-finite groups and their algebras 17

2.3 Polycyclic-by-finite groups and their algebras

In this section we state some important facts on polycyclic-by-finite groups and their group algebras. Basic references for such groups are [50, 64, 69].

The concept of a polycyclic-by-finite group yields a non-commutative generalization of a finitely generated abelian group.

Definition 2.3.1. A group G is said to be polycyclic-by-finite if it has a subnormal series

{1} = G₀⊆ G₁ ⊆ · · · ⊆ G_n−1⊆ G_n= G,

(so Gi / Gi+1, for 0 ≤ i ≤ n−1) such that G/Gn−1is finite and every other factor G_i/G_i−1 is cyclic. Equivalently, a group G is polycyclic-by-finite, if there exists a normal subgroup N of finite index in G that is either trivial or poly-infinite-cyclic, i.e. N has a subnormal series with infinite cyclic factors.

The number of infinite cyclic factors is independent of the choice of N . It is called the Hirsch rank of G and it is denoted by h(G).

If K is a field and G is an arbitrary group then the group algebra K[G] has an involution that K-linearly extends the classical involution on G which maps g to g⁻¹. It follows that K[G] is right Noetherian if and only if it is left Noetherian. Moreover, if K[G] is right Noetherian, then G satisfies the ascending chain condition on subgroups, in particular, G is finitely generated.

It remains an open problem to characterize when a group algebra is Noetherian. In case G is polycyclic-by-finite, an easy proof by induction on the Hirsch rank yields the following result ([64, Corollary 10.2.8]); it gives the only known class of groups for which the group algebra is Noetherian.

Theorem 2.3.2. Let K be a field. If G is a polycyclic-by-finite group then the group algebra K[G] is Noetherian.

For an arbitrary group G the set of torsion (we also say, periodic) elements is denoted by G⁺. If G⁺= {1} then G is said to be torsion-free.

It is well known that, if the group algebra K[G] is a domain, i.e. K[G] has no zero divisors, then G is torsion-free. Recall that the finite conjugacy centre (FC-centre for short) ∆(G) of a group G is the set of those elements g ∈ G that have only finitely many conjugates, or equivalently the centralizer C_G(g) of g in G has finite index. The set (∆(G))⁺ will be denoted as

∆⁺(G). Both sets are characteristic subgroups of G and ∆(G)/∆⁺(G) is

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2.3. Polycyclic-by-finite groups and their algebras 18

a torsion-free abelian group ([64, Lemma 4.1.6]). Clearly, a finite normal subgroup is contained in ∆⁺(G) and the latter group is trivial if and only if no such nontrivial subgroups exist. A group G so that ∆(G) = G is called a finite conjugacy group, or FC-group for short.

The following theorem states when the group algebra of a polycyclic- by-finite group is prime and when it is a domain (see [64, Theorem 4.2.10]

and [65, Theorem 37.5]).

Theorem 2.3.3. Let K be a field and G a polycyclic-by-finite group. Then the group algebra K[G] is prime if and only if ∆⁺(G) = {1}. Furthermore, the group algebra K[G] is a domain if and only if the group G is torsion-free.

We also mention the following result, which will be used often in this thesis and which can be found in ([60, Theorem 7.19]).

Theorem 2.3.4. Let K be a field. If S is a monoid with a group of right quotients G = SS⁻¹ then the semigroup algebra K[S] is prime if and only if K[G] is prime. Furthermore, K[S] is a domain if and only if K[G] is a domain.

Through the work of Roseblade (see for example [64, 65]), group algebras K[G] of polycyclic-by-finite groups G over a field K are rather well understood. In order to state the required properties some more notation and terminology is needed.

Suppose a group H acts on a set A. If a ∈ A and h ∈ H then we denote the action of h on a as a^h. An element a ∈ A is said to be H-orbital (or, simply, orbital) if its H-orbit is finite. The set of H-orbital elements in A we denote by ∆_H(A). Thus

∆H(A) = {a ∈ A | [H : CH(a)] < ∞}

with

CH(a) = {h ∈ H | a^h = a}.

Definition 2.3.5. Let H be a group. A finite dimensional Q[H]-module V is said to be a rational plinth for H if V is an irreducible Q[H1]-module for all subgroups H₁ of finite index in H. Now let H act on a finitely generated abelian group A. Then A is a plinth for H if, in additive notation, V = A ⊗_ZQ is a rational plinth.

Thus A is a plinth if and only if no proper pure subgroup of A is H- orbital. Recall that a subgroup B of A is pure if nA ∩ B = nB for all integers n. The plinth A is centric if and only if the centralizer C_H(A) is of finite index in H, or equivalently, A has rank one.

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2.4. Localization and Goldie’s theorem 19

A normal series

{1} = G₀⊆ G₁ ⊆ · · · ⊆ G_n= G

(every G_i/ G) for the polycyclic-by-finite group G is called a plinth series for G if each quotient Gi/Gi−1 is either finite or a plinth for G. It is not necessarily true that every G has a plinth series. However, any polycyclic- by-finite group G has a normal subgroup N of finite index with a plinth series, say

{1} = N₀⊆ N₁⊆ · · · ⊆ N_n= N.

The number of infinite factors N_i/N_i−1is called the plinth length of G and is denoted by pl(G). This parameter is independent of the choice of N and of the particular series for N . If F is a normal subgroup of G so that G/F is abelian then pl(G) ≤ pl(F ) + rk(G/F ) (recall that rk(B) denotes the torsion free rank of an abelian group B).

We now can formulate a fundamental result on prime ideals of group algebras of polycyclic-by-finite groups ([65, Theorem 19.6]).

Theorem 2.3.6. Let G be a polycyclic-by-finite group and K a field. The following properties hold.

1. clKdim(K[G]) = pl(G).

2. If K is absolute (that is, K is algebraic over a finite field) then every right primitive ideal M of K[G] is maximal and K[G]/M is finite dimensional.

We finish with one more result ([53]).

Theorem 2.3.7. A group algebra of a polycyclic-by-finite group is catenary.

2.4 Localization and Goldie’s theorem

We will now study Ore sets and localizations at prime ideals. We state Goldie’s theorem and we mention the behaviour of prime ideals under some localized rings. For more information about localization at prime ideals in non-commutative rings, we refer the reader to [33, 58].

For a subset X of a ring R (or more general for a subset X of a semigroup with zero) we put raR(X) = {r ∈ R | Xr = {0}}, the right annihilator of X in R. If X = {x} then we simply denote this set by raR(x). One says that an element x is right regular if ra_R(x) = {0}. Similarly one defines the

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left annihilator laR(X) and left regular elements. An element that is both right and left regular is said to be regular. Clearly, a non-regular element is a zero divisor and all elements in a domain are regular.

A multiplicatively closed subset X of a ring R is said to be a right Ore set if xR ∩ rX 6= ∅ for any x ∈ X and r ∈ R. If X consists of regular elements then RX⁻¹ denotes the ring of right quotients of R with respect to X. This ring will be called the localization of R with respect to X. If, moreover, X is the set consisting of all regular elements of R then RX⁻¹ is called the (classical) ring of right quotients of R and it is denoted by Qcl(R), or simply Q(R). So

Qcl(R) = {rx⁻¹| r, x ∈ R, x regular}.

In case the set of regular elements is a right Ore set one says that R has a classical ring of right quotients. The left versions of these notions are defined similarly.

Example 2.4.1. Let R be a commutative Noetherian ring and P a prime ideal of R. Then the set R \ P is an Ore set and the localization R(R \ P )⁻¹ is a local ring (that will be denoted by R_P) with unique maximal ideal P R_P. For non-commutative rings, much of this technique is not available.

Localization at a prime ideal can be impossible. Suppose R is a non- commutative ring. If P is a completely prime ideal, i.e. R/P is an integral domain, then R \ P is a multiplicatively closed (m.c. for short) subset of R. However for a general prime ideal P of R, R/P may have zero divisors and then R \ P is not a m.c. set. There is a natural m.c. set associated with an ideal I:

CR(I) = {s ∈ R | [s + I] is regular in R/I}.

Clearly, CR(0) is the set of all regular elements of R.

Definition 2.4.2. A prime ideal P of a Noetherian ring R is said to be localizable if CR(P ) is an Ore set. In this case, one can form the par- tial quotient ring R_P, by localizing at C_R(P ). The localization R_P is a Noetherian local ring with unique maximal ideal P R_P.

Although forming a non-commutative ring of fractions is not always possible, for Noetherian rings, Goldie’s theorem shows that the formation of a ring of fractions is always possible, not merely for an integral domain,

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but for any semiprime ring. Moreover, the ring of fractions is a semisimple Artinian ring. But before we state this theorem, we need some more definitions.

A ring R is said to satisfy the ascending chain condition on direct summands (of left ideals) if there do not exist strict ascending chains of the form

L1⊂ L₁⊕ L₂ ⊂ L₁⊕ L₂⊕ L₃ ⊂ · · ·

with every Li a left ideal of R. The (unique) number of direct summands is called the uniform dimension of R (or Goldie dimension). One says that a ring R satisfies the ascending chain condition on left annihilators if every ascending chain left annihilators stabilizes (the latter are left ideals of the form la_R(X) with X ⊆ R).

Definition 2.4.3. A ring R is (left) Goldie if R satisfies both the ascending chain condition on left annihilators and the ascending chain condition on direct summands of left ideals. Similarly one defines a right Goldie ring.

A ring that is both right and left Goldie is simply called a Goldie ring.

Clearly, a Noetherian ring is an example of a Goldie ring.

We are now ready for Goldie’ s theorem [58, Theorem 2.3.6]. Recall that a left ideal L of a ring R is said to be left essential if L ∩ X 6= {0} for any nonzero left ideal X of R.

Proposition 2.4.4. (Goldie) The following conditions are equivalent for a ring R.

1. R is a semiprime Goldie ring.

2. R has a classical ring of quotients which is semisimple Artinian.

3. A left ideal is essential if and only if it contains a regular element.

In particular, the ring R is prime Goldie if and only if R has a classical ring of quotients which is simple Artinian.

In particular, it follows from Goldie’ s theorem that a prime Noetherian ring has a simple Artinian classical ring of quotients. The following theorem, which follows immediately from the Wedderburn-Artin theorem (see for example [58, Theorem 0.1.10]), describes such simple Artinian rings of quotients.

Theorem 2.4.5. A ring R is a simple Artinian ring if and only if R ∼= Mn(D) for some (uniquely determined) natural number n and a division ring D.

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2.5. Maximal orders, Krull orders and class groups 22

To end this section, we consider the behaviour of prime ideals under some ring extensions. Again we only state one property that is relevant for later use (see [33, Theorem 9.20, Lemma 9.21, Theorem 9.22]).

Theorem 2.4.6. Let X be a right Ore set consisting of regular elements in a ring R. Assume that R is right Noetherian or that R satisfies a polynomial identity and RX⁻¹ is right Noetherian. Then the following properties hold.

1. If I is an ideal of R then I^e = IRX⁻¹ is an ideal of RX⁻¹.

2. The maps I → I^e and J → J ∩ R are inverse bijections between the set of prime ideals of RX⁻¹ and the set of those prime ideals of R that intersect X trivially.

2.5 Maximal orders, Krull orders and class groups

It is well known that integrally closed Noetherian domains, or more generally, Krull domains ([23]), are of fundamental importance in several areas of mathematics. In this section we concentrate on prime Noetherian maximal orders, Krull orders, Asano orders and unique factorization rings; these are generalizations to the class of noncommutative rings. In the vast literature one can find several types of such generalizations. Some of the relevant references for our purposes are [9, 10, 45, 52, 55, 56, 57, 71]. For a survey we refer the reader to [58].

Let R be a prime Goldie ring with classical ring of quotients Q = Q_cl(R).

One simply says that R is an order in Q. So, by Goldie’s theorem 2.4.4, every regular element of R is invertible in Q and elements in Q can be written as r₁c⁻¹₁ = c⁻¹₂ r₂, for some r₁, r₂, c₁, c₂ ∈ R with c₁, c₂ regular. If T is a subring of Q and a, b are invertible elements in Q so that aT b ⊆ R then also T is an order in Q. This leads to an equivalence relation on the orders in Q.

Definition 2.5.1. One says that two orders R₁ and R₂ in Q are equivalent if there are units a₁, a₂, b₁, b₂ ∈ Q so that a₁R₁b₁ ⊆ R₂ and a₂R₂b₂ ⊆ R₁. The order R is said to be maximal if it is maximal within its equivalence class.

A commutative integral domain D with field of fractions K is a maximal order if D is completely integrally closed, that is, if x, c ∈ K with c 6= 0 and cD[x] ⊆ D then x ∈ D. Clearly, a completely integrally closed domain

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is integrally closed , that is, if x ∈ K is so that D[x] is a finitely generated D-module then x ∈ D. If D is a Noetherian domain then both notions are equivalent.

The study of maximal orders is aided by the notion of a fractional ideal.

So suppose R is an order in its classical ring of quotients Q. A fractional R-ideal (or simply, fractional ideal if the order R is clear from the context) is a two-sided R-submodule I of Q so that aI ⊆ R and Ib ⊆ R for some invertible elements a, b ∈ Q. If, furthermore, I ⊆ R then I is called an integral fractional ideal. If A and B are subsets of Q then we put

(A :_r B) = {q ∈ Q | Bq ⊆ A}

and

(A :lB) = {q ∈ Q | qB ⊆ A}.

If I is a fractional R-ideal then (I :_rI) and (I :_lI) are orders in Q that are equivalent to R. We will often use the following characterization to check whether a ring is a maximal order.

Theorem 2.5.2. Suppose R is an order in Q. Then R is a maximal order if and only if R = (I :r I) = (I :l I) for every fractional R-ideal I (or equivalently, for every integral fractional R-ideal).

Assume now that R is a maximal order. It follows that (R :rI) = (R :l

I) for any fractional R-ideal I. One denotes this set simply as (R : I) or also as I⁻¹. Put I^∗ = (R : (R : I)), the divisorial closure of I. If I = I^∗ then I is said to be divisorial . Examples of such ideals are invertible fractional R-ideals J , that is J J⁰ = J⁰J = R, for some fractional R-ideal J⁰ (in this case, clearly, J⁰ = (R : J )). The following properties are readily verified for fractional R-ideals I and J ; I ⊆ I^∗, I^∗∗ = I^∗, (IJ )^∗ = (I^∗J^∗)^∗ = (I^∗J )^∗ = (IJ^∗)^∗, (R : I^∗) = (R : I) = (R : I)^∗, if I ⊆ J then I^∗⊆ J^∗. The divisorial product I ∗ J of two divisorial ideals I and J is defined as (IJ )^∗. It turns out that this product is commutative.

Definition 2.5.3. A prime Goldie ring R is said to be a Krull order if R is a maximal order that satisfies the ascending chain condition on divisorial integral ideals.

A prime Goldie ring R is said to be an Asano order if R is a maximal order with each ideal divisorial, or equivalent, each nonzero ideal of R is invertible.

Chatters and Jordan ([12]) have defined a unique factorization ring (UFR for short) to be a prime Noetherian ring R in which every nonzero

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prime ideal contains a prime ideal P generated by a normal element p, that is P = Rp = pR. Because of the principal ideal theorem 2.2.2, P is of height one. If, moreover, R and R/P have no zero divisors for each height one prime ideal P then R is said to be a unique factorization domain (UFD for short). The divisor group D(R) of a Krull order R is a free abelian group with basis the set of prime divisorial ideals. The latter are primes of height one and for rings satisfying a polynomial identity the height one primes are precisely the prime divisorial ideals. Furthermore, the normalizing class group of a Krull order, cl(R) is defined as D(R)/P (R), where P (R) is the subgroup consisting of the principal fractional ideals of R generated by a normal element. A Noetherian unique factorization ring is a Krull order with trivial normalizing class group ([12]).

In the next theorem we collect some of the essential properties of PI Krull orders. For details we refer the reader to [9, 10].

Theorem 2.5.4. Let R be a prime Krull order satisfying a polynomial identity. Then the following properties hold.

1. The divisorial ideals form a free abelian group with basis X¹(R), the height one primes of R.

2. If P ∈ X¹(R) then P ∩ Z(R) ∈ X¹(Z(R)), and furthermore, for any ideal I of R, I ⊆ P if and only if I ∩ Z(R) ⊆ P ∩ Z(R).

3. R = T R_Z(R)\P, where the intersection is taken over all height one primes of R, and every regular element r ∈ R is invertible in almost all (that is, except possibly finitely many) localizations RZ(R)\P. Fur- thermore, each R_Z(R)\P is a left and right principal ideal ring with a unique nonzero prime ideal.

4. For a multiplicatively closed set of ideals M of R, the (localized) ring RM = {q ∈ Qcl(R) | Iq ⊆ R, for some I ∈ M} is a Krull order, and

RM =\

RZ(R)\P,

where the intersection is taken over those height one primes P for which RM ⊆ R_Z(R)\P.

In the book of Reiner ([67]), the general theory of R-orders can be found. The ring R will always denote a Noetherian integral domain with a quotient field F . A will always be a finite dimensional F -algebra.

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Definition 2.5.5. An R-order in the F -algebra A is a subring Λ of A, having the same identity element as A, and such that Λ is a finitely generated module over R and generates A as a linear space over F . An order Λ is called maximal, if it is not properly contained in any other R-order.

From Theorem 2.5.4 and other results about PI-rings we get the following theorem, that gives a nice link between R- orders and the orders we considered before.

Theorem 2.5.6. The following conditions on a ring Λ with centre R are equivalent:

1. Λ is a prime PI ring;

2. Λ is an order in a central simple algebra;

3. Λ is an R-order in a central simple algebra.

In particular, if Λ is a PI domain, then Λ is an R-order in a central division algebra.

We will now give a characterization of group algebras of polycyclic-by- finite groups that are Noetherian prime maximal orders. These results are due to K.A. Brown and can be found in [6, 7]. It is also stated when such algebras are unique factorization rings or domains.

Recall that a group G is dihedral free if the normalizer of any subgroup H isomorphic with the infinite dihedral group D∞ = ha, b | b² = 1, ba = a⁻¹bi is of infinite index in G, that is, if H ∼= D∞ then H has infinitely many conjugates in G.

Theorem 2.5.7. Let G be a polycyclic-by-finite group and K a field. Then, K[G] is a Noetherian prime maximal order if and only if ∆⁺(G) = {1} and G is dihedral free.

Theorem 2.5.8. Let G be a polycyclic-by-finite group and K a field. Then, K[G] is a Noetherian unique factorization ring if and only if ∆⁺(G) = {1}, G is dihedral free and every plinth of G is centric.

From Theorem 2.5.7 it follows that the group algebra of a torsion-free polycyclic- by-finite group is a maximal order. Furthermore, from Theo- rem 2.5.8 and the following result it follows that the group algebra of a finitely generated torsion-free abelian-by-finite group is a unique factorization ring.

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Theorem 2.5.9. Let G be a polycyclic-by-finite group and K a field. If

∆⁺(G) = {1} then the following conditions are equivalent.

1. Every nonzero ideal of K[G] contains an invertible ideal.

2. Every nonzero ideal of K[G] contains a nonzero central element (this holds for example if K[G] is a PI algebra).

3. Every nonzero ideal of K[G] contains a nonzero normal element.

4. Every plinth of G is centric.

We end this chapter the way we started it, with semigroups. Definitions of orders and fractional ideals in the semigroup context are basically the same as in the ring case (see for example [28] and [71]). A cancellative monoid S which has a group of left and right quotients G is called an order. Such a monoid S is called a maximal order if there does not exist a submonoid S⁰ of G properly containing S and such that aS⁰b ⊆ S for some a, b ∈ G. An abelian cancellative monoid S is a maximal order in its group of quotients G if and only if it is completely integrally closed. The latter means that, if s, g ∈ G are such that {sgⁿ | n ∈ N0} ⊆ S then g ∈ S. In case S is finitely generated, S is completely integrally closed if and only if S is integrally closed, that is, if gⁿ ∈ S, with g ∈ G and some positive integer n, then g ∈ S.

For subsets A, B of G we define (A :_l B) = {g ∈ G | gB ⊆ A} and by (A :_r B) = {g ∈ G | Bg ⊆ A}. A f ractional ideal I of S means that SIS ⊆ I and cI, Id ⊆ S for some c, d ∈ S. If S is a maximal order, then (S :l I) = (S :r I) for any fractional ideal I; we simply denote this fractional ideal by (S : I) or by I⁻¹. Recall that then I is said to be d ivisorial if I = I^∗, where I^∗ = (S : (S : I)). A fractional ideal is said to be invertible if IJ = J I = S for some fractional ideal J of S. In this case J = I⁻¹ and I is a divisorial ideal. The divisorial product I ∗ J of two divisorial ideals I and J is defined as (IJ )^∗.

Theorem 2.5.2 yields necessary and sufficient conditions on the ideals for a ring R to be a maximal order. Furthermore it turns out that, for semigroups satisfying the ascending chain condition on two-sided ideals, conditions on the prime ideals are enough (see [45, Lemma 8.5.4]).

Lemma 2.5.10. Let S be a cancellative monoid which is an order (in its group of quotients G = SS⁻¹) and suppose S satisfies the ascending chain condition on two-sided ideals. Then S is a maximal order if and only if (P :_lP ) = (P :_rP ) = S for all P ∈ Spec(S).

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Similar as in the ring case, an order S is said to be a Krull order if and only if S is a maximal order satisfying the ascending chain condition on divisorial integral ideals (the latter are the fractional ideals contained in S).

In this case, the set D(S) of divisorial fractional ideals is a free abelian group for the ∗ operation. If S is a Krull order in a finitely generated abelian- by-finite group (so K[S] satisfies a polynomial identity by Theorem 2.2.5) then the minimal primes of S form a free basis for D(S). Furthermore, the normalizing class group of a Krull order S, cl(S), is defined as D(S)/P (S), where P (S) is the subgroup consisting of the principal fractional ideals of S generated by a normal element. In case the semigroup S is abelian we simply speak about the class group of S.

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Chapter 3 Maximal order semigroup algebras

In this chapter, we give necessary and sufficient conditions for a prime Noetherian algebra K[S] to be a maximal order, provided S is a submonoid of a polycyclic-by-finite group. Before we do this, we give a brief motivation why we focus on the semigroups under consideration and we state some known results on the topic. All the results in this chapter are joint work with Eric Jespers and Jan Okni´nski. The results in Section 3.2 and Section 3.3 have been published in [30] and the results in Section 3.4 will appear in [31].

3.1 Introduction and motivation

As said in Chapter 1, in general, it remains an unsolved problem to characterize when an arbitrary semigroup algebra is a prime Noetherian maximal order. Even for some concrete classes of finitely presented algebras it is not easily verified when these algebras are maximal orders. We state three very concrete examples.

Example 3.1.1. Let K be any field and let R = Khx₁, x₂, x₃, x₄i be the algebra defined by the following relations:

x1x4 = x2x3, x1x3 = x2x4, x3x1 = x4x2

x₃x₂= x₄x₁, x₁x₂= x₃x₄, x₂x₁ = x₄x₃. Is R a prime Noetherian maximal order?

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Noetherian semigroup algebras and prime maximal orders