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Author Horn, Larissa Dawn

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Fun with Tensor Products

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy

Mathematics

Larissa Dawn Horn

Committee in charge:

Professor Lance Small, Chair Professor Dan Rogalski Professor Paul Siegel Professor Jack Wolf Professor Eﬁm Zelmanov

Chair

University of California, San Diego

2008

iii DEDICATION

To all the friends, family, and loved ones who been here with me.

iv EPIGRAPH

In the sweetness of friendship let there be laughter, and sharing of pleasures. For in the dew of little things the heart ﬁnds its morning and is refreshed. —Kahlil Gibran

v TABLE OF CONTENTS

Signature Page ...... iii

Dedication ...... iv

Epigraph ...... v

Table of Contents ...... vi

Acknowledgements ...... vii

Vita and Publications ...... viii

Abstract ...... ix

1 Introduction ...... 1

2 Division Algebras ...... 5 2.1 PI Algebras ...... 5 2.2 Kaplansky’s Theorem ...... 6 2.3 A Second Proof ...... 7 2.4 The Algebraic Case ...... 8 2.5 The Transcendental Case ...... 9

3 The Martindale Ring of Quotients ...... 11 3.1 Deﬁnitions ...... 12 3.2 Examples of Centrally Closed Algebras ...... 13 3.3 Properties of Centrally Closed Algebras ...... 14 3.4 The Polynomial Ring Over a Centrally Closed Algebra ...... 15

4 Jacobson Rings ...... 16 4.1 Tensor Products with Simple Rings ...... 16 4.2 Tensoring with Centrally Closed Primitive Rings ...... 18

5 Krull Dimension ...... 19 5.1 Krull Dimension 0 ...... 20 5.2 Krull Dimension 1 ...... 21 5.3 Higher Krull Dimension and Other Questions ...... 23

Bibliography ...... 25

vi ACKNOWLEDGEMENTS

I would like to ﬁrst thank my advisor, Lance Small, for both mental and moral support over the past years. I would also like to thank Dan Rogalski for his many illustrative comments and John Farina for always being there to talk math and help translate Lance’s comments. And ﬁnally, I would like to thank all the students and faculty in the math department that make this such a welcoming place to learn alongside others. Not least amongst those are my dear friends. I thank Kristin for enjoying the sanity of the outdoors with me. I thank Amanda for the amazing times we have had when allowing a bit of insanity. And to the rest of the crowd, I am ever grateful for the conversations, the hikes, the bike rides, and the fellowship you have all given me. I thank my brother, Aaron, who helped me realize that there are many ways to be successful in life. And I thank Jon, who taught me that living a thoughtful life is one of them. I thank my parents, who gave me the independence and strong will to accomplish the things I desire. Finally, I thank Mike, who has made this year more wonderful than I ever expected. Your love means the world to me.

vii VITA

2003 B. S. in Mathematics magna cum laude, Emory University

2003-2008 Graduate Teaching Assistant, University of California, San Diego

2008 Ph. D. in Mathematics, University of California, San Diego

viii ABSTRACT OF THE DISSERTATION

Fun with Tensor Products

Larissa Dawn Horn Doctor of Philosophy in Mathematics

University of California San Diego, 2008

Professor Lance Small, Chair

This dissertation addresses various questions in noncommutative ring theory that may be solved with tensor product constructions. First, we consider division rings finite dimensional over their center. Using Ka- plansky’s Theorem one may show that any subdivision ring will also be finite dimensional over its center. We present an unpublished proof of this theorem, which generalizes to a new result: If D is a division ring algebraic over its center, Z, then any subdivision ring, D0, will also be algebraic over its center, Z0. In the following chapter we look at the Martindale ring of quotients of a ring. A ring, A, is called centrally closed if the center of its Martindale ring of quotients is equal to the center of A. If A is centrally closed, then any ideal of A ⊗k B will contain an ideal of the form I ⊗k J. We extend this result to show that prime ideals of A ⊗k B contain ideals of the form P ⊗k B or A ⊗k Q for P and Q prime ideals of A and B, respectively. Next, we consider Jacobson rings, and we show that a simple ring tensored with a Jacobson ring remains Jacobson. This determines new classes of Jacobson rings, broadening results of Jordan, Goodearl and Warfield. Finally, we look at the classical Krull dimension of polynomial rings over a noncommutative coefficient ring. We show that if R is centrally closed and has classical Krull dimension 1, then R[x] must have classical Krull dimension 2.

ix 1 Introduction

The idea of a tensor product arises naturally in algebra as a universal object in the category of multilinear maps of a ﬁxed set of modules. As we will see, there are many interesting applications of this construction. First, we look at some basic deﬁnitions and properties. In this paper, all rings will contain an identity element. Let R be a ring and

VR and RW right and left R-modules, respectively. For A, an abelian group, a bilinear map ϕ : V × W → A is said to be balanced if we have ϕ(va × w) = ϕ(v × aw) for all a ∈ R, v ∈ V , and w ∈ W .

We deﬁne the tensor product of V and W over R to be an abelian group V ⊗R W along with the balanced map i : V × W → V ⊗R W that satisﬁes the following universal criterion: For any balanced map from V × W to an abelian group, ϕ : V × W → A, there exists a unique homomorphism χ : V ⊗k W → A such that ϕ = χ ◦ i. That is, the following diagram commutes.

i V × W / V ⊗R W MMM MMM χ ϕ MM MMM & A

More concretely, we may deﬁne V ⊗RW as the quotient of a free group by certain relations. Set F = hV ×W i, the free abelian group on the set {(v, w)|v ∈ V and w ∈ W }. We will subsequently denote the element (v, w) ∈ F as v ⊗ w. Let H be the subgroup of F generated by the following elements:

• (v1 + v2) ⊗ w − v1 ⊗ w − v2 ⊗ w for all v1, v2 ∈ V and w ∈ W ,

• v ⊗ (w1 + w2) − v ⊗ w1 − v ⊗ w2 for all v ∈ V and w1, w2 ∈ W , and

• v ⊗ rw − vr ⊗ w for all v ∈ V , w ∈ W , and r ∈ R.

1 2

Then, we may define V ⊗R W to be the quotient F/H. If furthermore, our modules are algebras over a commutative ring or field, k, we may describe the tensor product as a k-algebra in the following way. For A and B k-algebras, we define multiplication in the tensor product by a1 ⊗b1 ·a2 ⊗b2 = a1a2 ⊗b1b2.

Since both A and B are k-algebras, α · a ⊗k b = αa ⊗k b = a ⊗k αb. In particular, for a simple ring, S, with center equal to the field k, we may tensor S over k with any other k-algebra. Note that if the field we are tensoring over is clear, we may drop the subscript and simply write A ⊗ B. Before we come to examples and properties, we should make a few comments on the tensor product construction. On the one hand, taking the free group on all ordered pairs from A and B gives a very large object; however, modding out by the relations may completely collapse the algebra. Understanding the resulting structure is often tricky and a very important part of algebra. There are but a few cases where we completely understand the structure of a tensor product. Some of the better understood examples of tensor products involve simple rings, polynomial rings, or matrix rings. For example, we can extend the coefficients of a polynomial ring by tensoring up to a larger ring. That is, if R embeds in S, then ∼ ∼ R[x] ⊗R S = S[x]. Similarly, for matrices over a k-algebra, A, we can write Mn(A) =

Mn(k) ⊗k A. In that tensor products often yield interesting objects of study, a natural question arises: How are structure-theoretic properties determined by the underlying rings? We would like to know which properties will pass through the construction. To this end, one of the most important problems is to ﬁnd constraints on the ideal structure of

A ⊗R B. Although I ⊗R J is an ideal of A ⊗R B whenever I and J are ideals of A and B, it is not generally true that all ideals are of this form. There are positive results only in certain cases. For central simple algebras the correspondence is perfect. An algebra, A, is central simple over a ﬁeld k if A is simple and the center of A, Z(A), is exactly k. If A is a central simple algebra over k and B is any other k-algebra, then the ideals of A ⊗k B are always of the form A ⊗k I for I an ideal of B. Similarly, prime ideals are of the form

A ⊗k P for P a prime ideal of B. For more on the theory of central simple algebras, see Chapter 4 of [Her94] or Jacobson [Jac64].

This implies that the ideals of a matrix ring Mn(A) are of the form Mn(I). 3

∼ Since Mn(k) is a simple ring with center of the form k · Idn = k, Mn(k) is central simple ∼ ∼ over k and so ideals of Mn(A) = Mn(k)⊗k A are of the form Mn(I) = Mn(k)⊗k I. These and other related properties of tensor products may be found in Chapter 3 of [Coh77]. Similar to the construction above extending the coefficients of a polynomial ring, we may extend scalars of a k-algebra. For R a k-algebra and K a field extension of k we produce a K-algebra by forming the tensor product R ⊗k K. In the tensor product, dimensions behave predictably in that dimkR = dimK (R ⊗k K). This useful fact will appear later in several proofs. The general theme of the results that follow is to use the well understood structure of simple rings to examine more complex problems. We will generalize results known for simple algebras to a broader class of rings. If we can then understand the structure of tensor products over these rings, we may prove useful results. In the first chapter, we will consider division rings finite dimensional over their center. Using Kaplansky’s Theorem one may show that any subdivision ring will also be finite dimensional over its center. This theorem was one of the first important applications of Polynomial Identity or PI theory. However, we will present an unpublished proof (attributed to Resco) of the same result, independent of PI theory. This proof will generalize to a new result: If D is a division ring algebraic over its center, Z, then any subdivision ring, D0, will also be algebraic over its center, Z0. In the second chapter we look at the Martindale ring of quotients of a ring. A ring, A, is called centrally closed if the center of its Martindale ring of quotients is equal to the center of A. A known result states that if A is a centrally closed, then any ideal of

A ⊗k B will contain a simple tensor and thus an ideal of the form I ⊗k J. We extend this result and a result of Lorenz in [Lor08] to show that prime ideals will, in fact, contain an ideal of the form P ⊗k B or A ⊗k Q for P and Q prime ideals of A and B, respectively. Next, we will consider Jacobson rings. These are an important class of rings that emerged from commutative algebraic geometry. Resco has proved that a simple ring tensored with a primitive ring remains primitive [Res80]. In this paper, we show that a simple ring tensored with a Jacobson ring is Jacobson. This identifies new classes of Jacobson rings, generalizing results of Jordan in [Jor75] and [Jor77] and Goodearl and Warfield in [GW82]. Finally, we look at the classical Krull dimension of polynomial rings over a noncommutative coefficient ring, R. The classical Krull dimension is the supremum of 4 lengths of chains of prime ideals. If the coefficient ring is commutative Noetherian, it is well known that the Krull dimension increases by exactly one when lifting to the polynomial ring. In the noncommutative case little progress has been made. However, we show that if R is centrally closed and has classical Krull dimension 1, then R[x] must have classical Krull dimension 2. 2 Division Algebras

One of the main techniques of ring theory is to consider rings and their ring extensions and the properties that pass between them. A division algebra, D, is a not necessarily commutative algebra in which every nonzero element is invertible. It is easily seen that the center of D will be a field. Thus, a division algebra is a central simple algebra over its center, Z. In this chapter, we consider a subdivision algebra of D, D0, and its center, Z0. It is natural to ask is whether there are dimensional properties that will pass from D to D0. We will begin by looking at the known result that if D is finite dimensional over Z, then D0 is finite dimensional over Z0. Looking at the case where D is algebraic over Z, we will show that D0 must also be algebraic over Z0.

2.1 PI Algebras

To understand the original proof of the finite dimensional case, we must first include some background on the theory of Polynomial Identity algebras or PI algebras. These algebras were first explicitly defined in [Kap48] where Kaplansky proves an important result in ring theory. They emerged as a generalization of commutative algebras and finite dimensional algebras. Notice that in a commutative ring, any two elements will satisfy the polynomial in noncommuting variables xy − yx. We define the standard polynomial of degree n to be the multilinear polynomial

sn(x1, x2, . . . , xn+1) = Σσ∈Sn+1 sgn(σ)xσ(1) ··· xσ(n+1) where again, the xi’s are noncommuting. If A is an n-dimensional algebra, any n + 1 elements of A are linearly dependent. Then, any n + 1 elements will always satisfy the standard polynomial of degree n + 1.

5 6

In this manner, we say that a k-algebra, A, satisﬁes a polynomial identity if there exists some nonzero polynomial f(x1, . . . , xn) ∈ khx1, . . . , xni with f(a1, . . . , an) =

0 for any ai ∈ A. We will refer to algebras that satisfy a polynomial identity as PI algebras. We have already seen that commutative and ﬁnite dimensional algebras are PI algebras. It is also clear that subalgebras and homomorphic images of PI algebras will be PI. We may also quickly see that the 2 × 2 matrices over a ﬁeld, k, satisfy the identity [[A, B]2,C] where brackets refer to the commutator. To see this, note that for any two matrices, A and B, the trace of [A, B] is zero. By the Cayley-Hamilton Theorem 2 X − tr(X)X + det(X) · Id2 = 0 for any 2 × 2 matrix, X. In particular, for [A, B] with 2 2 trace 0 we have [A, B] + det([A, B]) · Id2 = 0. Thus, [A, B] = −det([A, B]) · Id2 is a scalar matrix and so commutes with any other matrix. That is [[A, B]2,C] = 0 for any matrices A, B, C ∈ M2(k). The Amitsur-Levitzki Theorem gives us a precise answer for n × n matrices.

Mn(F ) satisfies the standard polynomial of degree 2n, but does not satisfy any polynomial of lower degree. There are many versions of this proof, including Amitsur and Levitzki’s original in [AL50] and Razmyslov and Rosset’s more recent proofs in [Raz74] and [Ros76]. PI theory emerged in considering various problems, in particular over commutative rings and finite dimensional algebras. However, it became clear that the class of PI algebras was much larger and that much deeper questions could be answered using the theory. For more background on PI theory and its development, see for example [Row80]. One of the first applications of PI theory was the use of Kaplansky’s Theorem to answer a question involving finite dimensional division algebras. This development highlighted the importance that PI theory would play in algebra over subsequent years. Below we present Kaplansky’s Theorem and show how it was used to solve an important problem in ring theory.

2.2 Kaplansky’s Theorem

If D is a ﬁnite dimensional algebra over its center Z, and D0 is a subdivision ring, then D0 must also be ﬁnite dimensional over its center, Z0. First we need to look at Kaplansky’s Theorem. 7

We say a ring, R, is right primitive if there exists a simple faithful right R- module. For example, all simple rings are primitive. In particular, division rings are always primitive. In the class of commutative rings, only fields are primitive. For a field is primitive by virtue of being simple, whereas if R is a primitive commutative ring it must have a maximal ideal, I such that R/I is a simple faithful module. But ann(R/I) = I and thus I = 0. That is, 0 is a maximal ideal, and so R is a field. Kaplansky’s Theorem gives a characterization of primitive PI rings as follows.

Theorem 2.2.1 (Kaplansky). A primitive ring satisfying a polynomial identity is ﬁnite dimensional over its center.

The proof of this theorem depends greatly on the Chevalley-Jacobson Density Theorem. A more thorough outline of the proof appears in [DF04]. One ﬁrst shows that

R must be isomorphic to Mn(D) for D a division ring. The result is then proven by showing that D is finite dimensional over its center. But now we return to the problem at hand. Let D be a division algebra finite dimensional over its center and D0 any sub-division algebra. Since D is finite dimensional over its center, D satisfies a polynomial identity. Then as a sub-division algebra, D0 also satisfies that same polynomial. Furthermore, we know that D0 is a primitive ring since it is a division algebra. Finally, applying Kaplansky’s Theorem to D0 we conclude that D0 is finite dimensional over its center. Hence we have the following theorem.

Theorem 2.2.2. Let D be a division algebra ﬁnite dimensional over its center, Z. Then any subdivision ring, D0, will also be ﬁnite dimensional over its center Z0.

Though this was one of the ﬁrst important applications of PI theory, around 1978 Herstein asked whether there existed an independent proof.

2.3 A Second Proof

About 25 years ago Resco suggested an answer to this question in an elegant and satisfyingly direct manner. Since there is, apparently, no published record of this proof, we present Resco’s argument below. It uses the compositum of two fields to reduce the problem to a tensor product construction. The compositum of two fields F and K is the smallest field in which both fields will embed. In the case that the two fields are contained in a larger division ring, L, and at least one of them is in the center of L, there 8 is a unique smallest field containing F and K still contained in L. We will denote the compositum of F and K as FK.

Proof of Theorem 2.2.2. Let D be a division algebra with center Z such that dimZ D < ∞. Consider D0 a sub division algebra of D with center Z0. Then the compositum ZZ0 is also a subﬁeld of D. 0 0 0 Note that by extension of scalars dimZ0 D = dimZZ0 (D ⊗Z0 ZZ ), and so we 0 0 reduce the problem to showing that dimZZ0 (D ⊗Z0 ZZ ) is ﬁnite. As a further reduc- 0 0 0 0 0 tion, consider the surjective ZZ -algebra homomorphism D ⊗Z0 ZZ → D (ZZ ) where D0(ZZ0) is the ring generated by D0 and ZZ0. Note that since D0 is central simple alge- 0 0 0 0 bra over Z , D ⊗Z0 ZZ is a simple ZZ -algebra. Hence the map is an isomorphism and 0 0 ∼ 0 0 D ⊗Z0 ZZ = D (ZZ ). We thus have the following diagram of inclusions:

D NN NNN D0(ZZ0) OO OOO D0

ZZ0 oo OOO ooo OOO Z o Z0

But as a subring of D, we know that D0(ZZ0) is ﬁnite dimensional over Z and hence also over ZZ0 containing Z. That is

0 0 0 0 0 dimZ0 D = dimZZ0 (D ⊗Z0 ZZ ) = dimZZ0 D (ZZ ) < ∞.

Thus, we have shown that D0 is ﬁnite dimensional over Z0.

Noting the purely algebraic constructions in this proof, one might ask if the technique will extend to other results. In the next section we consider one such situation, and in the last section we propose other scenarios where it might apply.

2.4 The Algebraic Case

Suppose now that D is algebraic over Z. That is, for every element d ∈ D, there is some f(x) ∈ Z[x] such that f(d) = 0. Note that this is equivalent to requiring 0 0 that dimZ Z[d] < ∞ for every d ∈ D. For a sub-division algebra, D , with center, Z , 9 we again consider dimensions over the compositum, ZZ0 to see that D0 is also algebraic over Z0.

Theorem 2.4.1. If D is algebraic over Z, then D0 will be algebraic over Z0.

Proof. Suppose D is algebraic over Z. Then to show that d0 ∈ D0 is algebraic over Z0, 0 0 we need to show that dimZ0 Z [d ] < ∞. But by extending coeﬃcients of the ring, we 0 0 0 0 0 0 0 ∼ 0 0 have dimZ0 Z [d ] = dimZZ0 Z [d ] ⊗Z0 ZZ . As before, D ⊗Z0 ZZ = D (ZZ ), and under 0 0 0 ∼ 0 0 this isomorphism, Z [d ] ⊗Z0 ZZ = ZZ [d ]. So we reduce the problem to showing that d0 is algebraic over ZZ0. The diagram below summarizes the situation.

D VVVV VVVV VVVV D0(ZZ0) U oo UUUU oo UUUU ZZ0[d0] U D0 @ w @@ www @@ 0 0 @@ Z [d ] @@ 6 @ 66 ZZ0 66 ggg VVVV 6 gggg VVVV 66 gggg VVVV Z ggg V Z0 Since d0 ∈ D we know that it is algebraic over Z. That is, d0 satisﬁes a polynomial f(x) ∈ Z[x]. But as Z[x] ⊆ ZZ0[x], d0 also satisﬁes a polynomial in ZZ0[x] and thus is algebraic over ZZ0 as desired.

2.5 The Transcendental Case

A first remark is that these theorems are perhaps more specialized than necessary. We really only needed the centers of our rings to be fields and for the compositum to still lie in the original ring. With these two conditions on a ring, both the finite dimensional and algebraic proofs will go through. So we might ask what hypotheses will satisfy this criterion. We also might hope to extend this result even further to questions involving other dimensional properties. For example, suppose the transcendence degrees of maximal commutative subfields of D lying over its center are bounded. Will this bound or some other bound extend to subdivision rings? Another interesting dimensional property of algebras is the Gelfand-Kirillov or GK dimension. This dimension characterizes the growth of finite dimensional subspaces 10 of infinite dimensional algebras. Suppose D is a division algebra with center Z. Take V a finite dimensional Z-subspace of D such that 1 ∈ V and V generates D as an algebra.

We deﬁne dV (n) as the dimension over Z of the subspace of D spanned by products of n elements from V . Then we deﬁne

GKdimZ D = limn→∞ logn dV (n).

In fact, it can be shown that the GK dimension will be independent of choice of V . More generally, if D is not ﬁnitely generated, we deﬁne

0 0 GKdimZ D = sup{GKdim(D )|D ⊆ D,D ﬁnitely generated}.

For example, algebras of GK dimension 0 are simply the ﬁnite dimensional algebras. The argument in this chapter may extend to show that the GK dimension of D will bound the GK dimension of any subalgebra over its center. 3 The Martindale Ring of Quotients

The idea of a ring of quotients for noncommutative rings has been an alluring prospect for quite some time. Many commutative ring theory proofs rely on the theory of localization in critical steps. In order to generalize these theorems, we often would like to use noncommutative localization. For a ring, A, and a multiplicative subset of regular elements, S, we say that an embedding of A in Q is a localization of A at S if every element of S is invertible in Q. Unfortunately, the question of when we are able to invert elements in a noncommutative ring is not as simple as it might seem. In fact, in [Mal37] Malcev gives an example of a domain that cannot be em- bedded in a division ring. Take Qha, b, c, d, e, f, g, hi the free algebra on eight generators. Let H be the ideal generated by the relations ae = bf, ce = df, and ag = bh. Then the algebra Qha, b, c, d, e, f, g, hi/H is a domain, but if it were contained in a division ring, we must have cg = dh. However, this relation does not hold in Qha, b, c, d, e, f, g, hi/H. If we require additionally that each element of the localization be expressible as ab−1 for a ∈ A and b ∈ S, then we have some significant results. First, Ore proved that this requirement is equivalent to having common right multiples. That is, given a ∈ R and s ∈ S, there exist a0 ∈ R and s0 ∈ S such that as0 = sa0. This condition is referred to as the Ore condition. A ring is right localizable in the sense of Ore if and only if it satisfies the Ore condition. Thus, to form the right Ore rings of quotients, we need only check when rings satisfy the Ore condition. An important step towards this was made with Goldie’s Theorem, which gives us not only a large class of rings satisfying the Ore condition but also describes the localization of such rings. We say a ring is right Goldie if it satisfies the ascending chain

11 12 condition on right annihilators and contains no inﬁnite direct sum of right ideals. In particular, a Noetherian ring will always be Goldie. Goldie’s Theorem states that a ring R has a semisimple right Artinian quotient ring if and only if R is semiprime right Goldie. Additionally, R has a simple right Artinian quotient ring if and only if R is prime right Goldie. An excellent reference for Ore localization and Goldie’s Theorem may be found in Chapter 7 of [Her94]. Another important construction is Martindale’s ring of quotients. In [Mar69], Martindale shows that we may embed any prime ring in a ring in which all central elements are invertible. We will see that there are several important properties of this ring of quotients. Using these properties we generalize results, which formerly depended on chain conditions and other hypotheses, to require only the prime hypothesis.

3.1 Deﬁnitions

Given a prime ring A, we construct the right Martindale ring of quotients as follows. Consider the set {[I, f]} where I is a nonzero ideal of A and f : I → A is a right A-module map. Now deﬁne an equivalence on the set by [I, f] ∼ [J, g] if and only if there exists some nonzero ideal L ⊆ I ∩J with f|L = g|L. We call the ring Qr(A) = {[I, f]}/ ∼ the right Martindale ring of quotients. On this set we have addition

[I, f] + [J, g] = [I ∩ J, f + g] and multiplication [I, f] · [J, g] = [JI, f ◦ g].

Note that these operations will preserve the equivalence relation by intersecting down to smaller ideals if necessary. Also, these operations are well-deﬁned, since if A is a prime ring, two nonzero ideals cannot have trivial intersection. For if I ∩ J = (0), then IJ ⊆ I ∩ J = (0). So, by the primeness of A either I = (0) or J = (0). Then, I ∩ J is again a nontrivial ideal of A. Similarly, the ideal JI will be nonzero. For the deﬁnition of multiplication, we use the ideal JI so that the composition f ◦ g makes sense. For then, g(JI) ⊆ g(J) · I ⊆ I.

In fact, A,→ Qr(A) via a 7→ [A, λa] where λa is left multiplication by a.

Furthermore, we have an injection since [A, λa] = 0 implies that aI = (0) for some 13 nonzero ideal I of A. But then (a)I = AaAI ⊆ AaI = 0 and the primeness of A would imply that I = (0) or a = 0. Additionally, central elements will be invertible. For if a is a central element, then aA is a two-sided ideal, and so [aA, ax 7→ x] is an inverse to [A, λa]. For more details on the construction of the Martindale ring of quotients, see Section 3.4 in [Row88] or [BFPR08]. Of the rings this construction applies to, perhaps most important are the centrally closed rings. We deﬁne the extended center of A to be the center of the Martindale ring of quotients. A ring is called centrally closed if its center is equal to the extended center. Centrally closed rings have many properties similar to central simple algebras as we will see in the third section of this chapter. First, we examine several examples to see that many rings are, in fact, centrally closed.

3.2 Examples of Centrally Closed Algebras

Example. A simple ring is centrally closed. Note that the center of the ring of quotients is precisely the set of [I, f] such that f : I → A is an (A, A)-module homomorphism. If A is simple, then the only bimodule homomorphisms f : A → A are right multiplication by a central element. Thus, the center of the quotient ring is precisely the center of A.

Example. By Theorem 1 in [RS97], an aﬃne primitive Noetherian ring over an uncountable algebraically closed ﬁeld is centrally closed. For example, if q is a nonzero complex number that is not a root of unity, the quantum plane over C or Chx, y|xy = qyxi is centrally closed.

Let An(k) be the algebra generated by x1, x2, . . . , xn, y1, y2, . . . , yn with the relations xiyj = yjxi for i 6= j and xiyj − yjxi = 1 for i = j. This algebra is called the nth Weyl algebra. In particular, the ﬁrst Weyl algebra is just khx, y|xy − yx = 1i. In the ﬁrst Weyl algebra, the idealizer of the right ideal xA1(k) will be a centrally closed algebra.

Example. A k-algebra is called just infinite if dimk A = ∞, but for every two sided ideal, I, dimk A/I < ∞. In [BFPR08], Bell, Farina, and Pendergrass-Rice show that if A is a right Noetherian just infinite algebra over an algebraically closed field and A does not satisfy a polynomial identity, then A must be centrally closed.

Example. The following interesting example was pointed out by Small. 14

Consider the centerless Virasoro algebra, W . This is deﬁned to be the Lie algebra generated over C by {ei}i∈Z with multiplication deﬁned by [ei, ej] = (j − i)ei+j. The universal enveloping algebra of the centerless Virasoro algebra is an example of a centrally closed algebra that is not simple. In fact, it is not known whether U(W ) is even Noetherian. In an unpublished result, Wallach has shown that the universal enveloping algebra of this Lie algebra over C is primitive. Since W has GK dimension 1, [Smi76] shows that U(W ) has subexponential growth. Then, noting that [Jat69] shows that algebras with subexponential growth are localizable in the sense of Ore, set D = Q(U(W )).

In [Oom74], Ooms describes the embedding Z(D) → EndU(W )V for any faithful irreducible U(W )-module, V . By Schur’s Lemma, EndU(W )V is a division ring. Then, since U(W ) is of countable dimension over an uncountable ﬁeld, Amitsur’s trick gives us that U(W ) satisﬁes the Nullstellensatz. This implies that EndU V is algebraic over C and thus equal to C. Then, by the embedding we have that Z(D) = C and we have shown that U(W ) is centrally closed.

3.3 Properties of Centrally Closed Algebras

As mentioned, centrally closed rings are similar to simple rings in many ways. Note that the center of a centrally closed algebra is always a ﬁeld as in the case of simple rings. Furthermore, the tensor product of a centrally closed algebra with a ﬁeld remains centrally closed. Most importantly, in [EMO75] we have the following theorem regarding ideals of tensor products.

Theorem 3.3.1 (Martindale, Erickson, Osborn). If R is a centrally closed algebra over k, and S is any other k-algebra, then an ideal of R ⊗k S contains a nonzero ideal of the form I ⊗k J for I an ideal of R and J an ideal of S.

Though this gives us some idea of the ideal structure of the tensor product, for prime ideals we can say a bit more about the ideal I ⊗k J.

Theorem 3.3.2. If R is a centrally closed algebra over k and S any other k-algebra, then any prime ideal of R ⊗k S contains an ideal of the form I ⊗k J for I a nonzero prime ideal of R and J a nonzero prime ideal of S.

Proof. Let P C R ⊗k S be a prime ideal. By Theorem 3.3.1, we know that P contains an ideal of the form I ⊗k J where I is an ideal of R and J is an ideal of S. 15

0 0 Now, extend I to the largest ideal of R, I such that I ⊗k J ⊆ P . We will 0 0 show that I is in fact a prime ideal. For if I1 and I2 are ideals of R and I1I2 ⊆ I , then 0 (I1 ⊗k J)(I2 ⊗k J) ⊆ I ⊗k J ⊆ P . So, either I1 ⊗k J ⊆ P or I2 ⊗k J ⊆ P . In the ﬁrst 0 0 0 0 case, by the maximality of I , I1 ⊆ I and similarly in the latter I2 ⊆ I . Thus, I must be a prime ideal of R. In the same way, we take J 0 to be the maximal ideal such that I0 ⊗ J 0 ⊆ P . Again we ﬁnd that, J 0 will be prime.

Now, using the fact that P is a prime ideal, we take this one step further.

Corollary 3.3.3. If R is a centrally closed algebra over k and S any other k-algebra, then any prime ideal of R ⊗k S contains an ideal of the form I ⊗k S or R ⊗k J for I a nonzero prime ideal of R and J a nonzero prime ideal of S.

Proof. By the preceding theorem, if P C R is a prime ideal, it must contain an ideal of the form I ⊗k J for I a prime ideal of R and J a prime ideal of S. Now, since

(I ⊗k S)(R ⊗k J) = I ⊗k J ⊆ P and P is a prime ideal, then either I ⊗k S ⊆ P or

R ⊗k J ⊆ P as desired.

In [Lor08], Martin Lorenz examines similar ideas and shows further that if

P C R ⊗k S is a prime ideal, and P ∩ R = 0, then P = R ⊗k (P ∩ S).

3.4 The Polynomial Ring Over a Centrally Closed Algebra

In Corollary 3.3.3, we have managed to capture a property of a prime ideal in a tensor product in terms of prime ideals of R and S. We are particularly interested in the case where S = k[x] and R = A is a centrally closed k-algebra. For in this case, we ∼ know R ⊗k S = A[x], the polynomial ring with coeﬃcients in A. Corollary 3.3.3 along with Lorenz’s result then implies that any prime ideal of A[x] either contains an ideal of the form Q[x] for Q a nonzero prime ideal of A or is of the form (q) where q ∈ k[x] is a nonzero central element. But since k[x] is just a commutative polynomial ring over a ﬁeld and (q) is a prime ideal, q must be irreducible. If further, k is algebraically closed, q is of the form x − α for some α ∈ k. That is,

Theorem 3.4.1. If A is a centrally closed algebra over an algebraically closed ﬁeld, k, and (0) 6= P C A[x] is a prime ideal, then P contains Q[x] for Q a nonzero prime ideal of A or P is the ideal generated by x − α for α ∈ k. 4 Jacobson Rings

We begin with some deﬁnitions. Recall that a ring, R, is called primitive if there exists a simple faithful R-module. An ideal, I is primitive if R/I is a primitive ring. A ring is called semiprimitive if (0) is an intersection of primitive ideals. We deﬁne the Jacobson radical to be the intersection of all primitive ideals in R and denote it J(R). A ring is called Jacobson if J(R/P ) = (0) for every prime ideal P ⊆ R. Equivalently, a ring is Jacobson if and only if every prime image is semiprimitive. That is every prime ideal is the intersection of primitive ideals. The idea of Jacobson rings can be traced to papers of Jacobson, Krull and Goldman in the 1950s where they were looking at rings satisfying the conditions of the commutative Nullstellensatz. See for example [Jac45], [Gol51], and [Kru51].

4.1 Tensor Products with Simple Rings

As mentioned, the idea of Jacobson rings was motivated by an interesting analog in the commutative case. The deﬁnition arose in studying when rings satisfy conditions of a commutative version of the Nullstellensatz, which states that a ﬁnitely generated extension of a Jacobson ring is still Jacobson. Thus, it is of interest to know which ring extensions are Jacobson also in the noncommutative setting. We begin with a remark of Small that can be found in [Res80]. First, note that primitivity is equivalent to co-maximality of some one-sided ideal. That is, a ring, A, is primitive if and only if there exists a proper one-sided ideal, M, such that M + I = A for every two sided ideal, I.

Lemma 4.1.1 (Small). Let S be a central simple algebra over a ﬁeld, k. Then if A is a primitive k-algebra, we have A ⊗k S is also primitive.

Proof. Since A is primitive, there is some right ideal, M, such that M + I = A for

16 17

every ideal I C A. Then, we will show the right ideal M ⊗k S of A ⊗k S satisﬁes the co-maximality condition. For any ideal of A ⊗k S must be of the form I ⊗k S. Then,

(M ⊗k S) + (I ⊗k S) = (M + I) ⊗k S = A ⊗k S.

We use this well-behaved property of simple algebras in tensor products to prove the following result.

Theorem 4.1.2. If J is a Jacobson algebra over a ﬁeld k and S a simple ring with center k, then J ⊗k S is still a Jacobson algebra over k.

Proof. Note that J ⊗k S is Jacobson if and only if the Jacobson radical of every prime image is 0. Since S is central simple, prime ideals of J ⊗k S are of the form P ⊗k S for P ∼ a prime ideal of J. Then prime images are of the form (J ⊗k S)/(P ⊗k S) = (J/P ) ⊗k S. By Lemma 4.1.1, if Q is a primitive k-algebra and S a central simple k-algebra, then Q ⊗k S is primitive. Then, for each primitive ideal Qi ⊆ J/P , we have Qi ⊗k S is also a primitive ideal of (J/P ) ⊗k S.

Then, J((J/P )⊗k S), being the intersection of all primitive ideals of (J/P )⊗k S, must be contained in ∩(Qi ⊗k S) where Qi runs over all primitive ideals of J/P . Since

J is Jacobson, we have ∩(Qi ⊗k S) = (∩Qi) ⊗k S = J(J/P ) ⊗k S = (0) ⊗k S = 0, that is

J((J/P ) ⊗k S) ⊆ (0).

Then we have shown that an arbitrary prime image of J ⊗k S has Jacobson radical equal to 0. Thus, J ⊗k S is a Jacobson ring.

This theorem allows us to broaden a result of Jordan, constructing a new set of interesting examples of Jacobson rings. We look at a certain set of skew polynomial rings. Recall that δ is a derivation on a ring J if δ(rs) = rδ(s) + δ(r)s. Let J be a k-algebra of characteristic 0 and δ a derivation of J. We construct the differential operator ring, J[x, δ], as follows. Take the set of polynomials in x with coefficients in J where the coefficients do not necessarily commute with the indeterminate. Instead, commutation is governed by the relation rx = xr+δ(r) for r ∈ J. For example, J[y][x, δ] where δ = d/dy yields the first Weyl algebra over J. In [Jor75] Jordan proves that J[x, δ] is Jacobson whenever J is a Noetherian Jacobson ring. He further shows that this implies that the Weyl algebra over a Noetherian

Jacobson algebra of characteristic 0, A1(J), is Jacobson. 18

However, we now see that the Noetherian hypothesis is unnecessary. For we ∼ ∼ have J[y][x, δ] = J ⊗k k[y][x, δ] = J ⊗k A1(k). Since in characteristic 0, A1(k) is a simple ring with center k, Theorem 4.1.2 shows that the tensor product J ⊗k A1(k) is also Jacobson. And we have the following corollary.

Corollary 4.1.3. Let J be a Jacobson algebra over a ﬁeld of characteristic 0. Then the ﬁrst Weyl algebra over J is Jacobson.

Note that the requirement of characteristic 0 is necessary for Jordan also gives an example of a Jacobson algebra over a ﬁeld of nonzero characteristic where this result does not hold. In fact, we may see that our proof breaks down precisely because the Weyl algebra is not simple over a ﬁeld of nonzero characteristic.

4.2 Tensoring with Centrally Closed Primitive Rings

Also in [Res80], Resco shows a nearly analogous result for tensor products of centrally closed primitive algebras with primitive algebras.

Theorem 4.2.1 (Resco). Let C be a centrally closed primitive algebra over a ﬁeld, k.

Then if A is a primitive k-algebra, A ⊗k C is also primitive.

We might consider whether this would imply that a centrally closed primitive ring tensored with a Jacobson ring is still Jacobson, which would extend our set of examples. 5 Krull Dimension

We define the height of a prime ideal, P to be the supremum of lengths of chains of prime ideals P0 ⊆ P1 ⊆ · · · ⊆ Pn = P where the length counts the number of jumps, e.g. the length of P0 ⊆ P1 is 1. The classical Krull dimension of a ring is defined to be the supremum of the lengths of chains of prime ideals. We will denote this as k.dim A. For commutative rings, Krull dimension corresponds to a geometric dimension in algebraic geometry. As always, we would like to see this transfer nicely to the noncommutative setting, but GK dimension turns out to be a more useful measure of dimension. However, there are other instances where we might hope to find useful results.

A polynomial ring over a commutative Noetherian ring, R[x1, . . . , xn], has classical Krull dimension equal to the Krull dimension of R plus the number of indetermi- nates. Many proofs of this exist in the literature, for example Eisenbud’s [Eis95]. In general, very little progress has been made regarding the classical Krull dimension of polynomial rings in the noncommutative case. For a Noetherian PI ring, A we know that k.dim A[x] = k.dim A + 1. See for example, [GW82]. We would like this to be true for general Noetherian rings. To begin looking at this problem, we consider a few properties of prime ideals in polynomial rings that will be useful in what follows. First we claim that if P C A is prime, then P [x] C A[x] is also prime. By modding out by P [x] it suﬃces to show that if n A is a prime ring, then A[x] is also. Suppose 0 6= a, b ∈ A[x] with a = anx +...+a1x+a0 m and b = bmx + ... + b1x + b0 where an, bm 6= 0. Then, if aA[x]b = 0, we also have n m aAb = 0. Taking the highest degree term, anx Abmx = 0, or anAbm = 0. But since

A is prime, this implies that either an = 0 or bm = 0, which is a contradiction to our assumption. Hence, either a = 0 or b = 0 and A[x] is prime. Note that this implies that if k.dim A = n, then k.dim A[x] ≥ n+1 for any ring,

A. For given a chain of prime ideals of length n in A, we have P0[x] ⊆ P2[x] ⊆ · · · ⊆ Pn[x]

19 20

is a chain of primes in A[x]. Note however, Pn[x] is not maximal, for Pn[x] ⊆ (Pn[x], x) for example. Then we have found a chain of length at least n + 1 and so k.dim A[x] ≥ n + 1. Additionally, if P C A[x] is prime, then Pe = P ∩ A C A is also prime. For, suppose I and J are ideals of A such that IJ ⊆ Pe. Then we have I[x]J[x] ⊆ Pe[x] ⊆ P . Then either I[x] ⊆ P or J[x] ⊆ P . In the ﬁrst case, we have I ⊆ Pe and in the latter J ⊆ Pe. We will need one other fact about prime ideals. Given a ring, A, and an Ore set, S, with prime ideals, I and J, such that I ∩ S = ∅ and J ∩ S = ∅. Then if I ( J, −1 −1 we have IS ( JS , i.e. when localizing away from prime ideals proper containment is preserved.

5.1 Krull Dimension 0

To begin, we look at a simple ring, that is a prime ring of classical Krull dimension 0. In this case, we have the following.

Proposition 5.1.1. If S is a simple ring, then k.dim S[x1, x2, . . . , xn] = n.

Proof. Let k be the center of S. Then k is a ﬁeld, and S is central simple over k. We form the tensor product S ⊗k k[x1, x2, . . . , xn] which is isomorphic to S[x1, x2, . . . , xn].

But, the prime ideals of S ⊗k k[x1, x2, . . . , xn] must be of the form S ⊗k P for P a prime ideal of k[x1, x2, . . . , xn]. Hence,

k.dim S[x1, x2, . . . , xn] = k.dim S ⊗k k[x1, x2, . . . , xn] = k.dim k[x1, x2, . . . , xn].

Since k is a commutative Noetherian ring we have already noted k.dim k[x1, x2, . . . , xn] = n. Thus k.dim S[x1, x2, . . . , xn] = n.

We can use Proposition 5.1.1 to prove the following important lemma. The lemma will then allow us to consider chains of prime ideals in A[x] more carefully.

Lemma 5.1.2. Let A be a Noetherian ring and suppose we have properly contained ideals P0 ( P1 ( A[x]. If P0 ∩ A = P1 ∩ A = Pe, then P0 = Pe[x].

Proof. Suppose that we have P0 ∩ A = P1 ∩ A = Pe. By way of contradiction suppose that P0 6= Pe[x]. Now, it is clear that Pe[x] ⊆ P0. So, we have the chain of properly contained prime ideals

Pe[x] ( P0 ( P1 ( A[x]. 21

Modding out by Pe[x] reduces to the chain

P0 P1 A[x] 0 ( ( ( . Pe[x] Pe[x] Pe[x] A[x] A A Notice that =∼ [x]. Then is prime Noetherian ring, and hence prime Pe[x] Pe Pe A Goldie. So, by Goldie’s theorem we may localize [x] at T , where T is the set of regular Pe A elements of . The quotient will be a polynomial ring over a simple Artinian ring, S[x]. Pe A A + Pe[x] Under the isomorphism, corresponds to . Further we have that each of our Pe Pe[x] ideals intersects T trivially. For,

P0 A + Pe[x] P0 ∩ (A + Pe[x]) Pe[x] ∩ = = = 0, Pe[x] Pe[x] Pe[x] Pe[x] and P1 A + Pe[x] P1 ∩ (A + Pe[x]) Pe[x] ∩ = = = 0. Pe[x] Pe[x] Pe[x] Pe[x] P P So, localization at T preserves containment. Letting the localizations of 0 and 1 Pe[x] Pe[x] be Q0 and Q1, respectively, we have

0 ( Q0 ( Q1 ( S[x] which is impossible since k.dim S[x] = 1. Consequently, we must have P0 = Pe[x].

For ease of notation, we will subsequently denote Pi ∩ A = Pei. Note, the preceding lemma implies that for a Noetherian ring, A, you cannot have 3 properly contained ideals in A[x] which intersect to the same ideal of A. For, if we have Pi ( Pi+1 ( Pi+2 and Pei = Pgi+1 = Pgi+2, then by Lemma 5.1.2, Pi+1 = Pgi+1[x] and Pi = Pei[x] and so, Pi = Pi+1.

5.2 Krull Dimension 1

Jategaonkar’s Principal Ideal Theorem, which can be found in Section 4.1.11 of [MR01], gives a criterion for primes to be height 1. We present a specialized instance below. The proof is a standard proof, considered to be folklore.

Lemma 5.2.1. For A a Noetherian ring, let q ∈ k[x] be a nonzero irreducible central element in A[x]. Then (q) has height 1. 22

Proof. For simplicity of notation, set A[x] = R. Now, suppose Q is a prime ideal such that Q ( (q). Set I = {r ∈ R|qr ∈ Q}. With this definition, it is clear that Q ⊇ qI. Then, Q ⊆ (q) implies that any element x ∈ Q can be written as x = qr. So, r ∈ I by definition, and we have Q ⊆ qI. Then Q = qI. But Q is a prime ideal and so (q)I ⊆ Q implies that either (q) or I is contained in Q. The first is impossible, so the latter must hold. Then, Q = qI ⊆ qQ ⊆ Q and so Q = qQ. Replacing the second Q with qQ we have Q = q2Q and so on. That is, Q = qnQ for all n. This gives us Q ⊆ ∩qnQ ⊆ ∩qnR. But ∩qnR = (0) must be zero. For if it were nonzero, then it would contain a regular element 0 6= c ∈ ∩qnR, 2 2 then c = qr1 = q r2 = ... or since q is central, c = r1q = r2q = .... Note that since q ∈ k[x], q is regular. Canceling q we have r1 = r2q and in general ri = ri+1q. So, each ri is in ri+1R. We form the ascending chain r1R ⊆ r2R ⊆ r3R ⊆ · · · which must terminate by the Noetherian condition. That is, for some i, riR = ri+1R. In particular, ri+1 = ris for some s ∈ R. Now, since c and q are regular, ri must also be regular for all i. Then i i+1 i+1 c = q ri = q ri+1 = q ris or 1 = qs. But then (q) = R, which is a contradiction. Thus, Q ⊆ ∩qnR = (0), as desired. So we have shown that the only ideal properly contained in (q) is (0). Then (q) has height 1.

We wish to show that for a prime Noetherian centrally closed algebra of classical Krull dimension 1 over an algebraically closed ﬁeld, the dimension increases by exactly one when extending to the polynomial ring. To do this, we bring together several ideas thus far developed. First, we use the description of the prime ideals from Theorem 3.4.1. As we noted in the previous section, there do not exist chains of prime ideals

Pi ( Pi+1 ( Pi+2 ( A[x] where all prime ideals have the same intersection with A. We use this idea again to examine the possibilities when k.dim A = 1.

Proposition 5.2.2. Suppose A is a prime Noetherian centrally closed algebra with center, k, which is algebraically closed. If the classical Krull dimension of A is 1, then the classical Krull dimension of A[x] is 2.

Proof. We will prove this in two steps. Given a prime P ∈ A[x], ﬁrst consider the case where P ∩ A = 0. By Theorem 3.4.1 P must equal (x − α) for some α ∈ k. Now since A[x]/(x − α) =∼ A, a prime ring, (x − α) is a prime ideal. Then by Lemma 5.2.1, we see that (x−α) has height one. Now, for a prime ideal, Q, k.dim A[x] = htQ+k.dim A[x]/Q. Taking Q = (x−α), gives k.dim A[x] = 1+k.dim A. Since A has classical Krull dimension 1, we have shown k.dim A[x] = 2. 23

Now, we may assume that P ∩A 6= 0 for all nontrivial primes P . Suppose there were a chain

0 ( P1 ( P2 ( P3 ( A[x].

Intersecting with A yields

0 ( Pf1 ⊆ Pf2 ⊆ Pf3 ( A.

But, there can be only one nontrivial prime in a chain of prime ideals in A. Hence

Pf1 = Pf2 = Pf3.

Then, Lemma 5.1.2 implies that Pf1[x] = P1 and Pf2[x] = P2. Thus we have shown that

P1 = P2. In A[x] we can have a chain of at most 2 nonzero prime ideals. Recall that for any ring, A, k.dim A[x] ≥ k.dim A + 1. So, in this case, the Krull dimension of A[x] is exactly 2.

5.3 Higher Krull Dimension and Other Questions

We should be able to extend this technique to show that the classical Krull dimension of any prime Noetherian centrally closed algebra increases by exactly one when forming the polynomial ring. Note that for the ﬁrst part of the proof of Proposition 5.2.2 we need not assume Krull dimension 1. If any prime has nontrivial intersection with A, then k.dim A[x] = 1 + k.dim A. So, we may reduce the question to the case where every prime intersects A nontrivially. In this case, we have also shown that the classical Krull dimension of a polynomial ring over any prime Noetherian ring can be at most equal to 2 · k.dim A + 1.

For we may take a chain of prime ideals P0 ( P1 ( ··· ( Pn in A[x] and then consider the intersection of this chain with the coeﬃcient ring. By the remarks following Lemma 5.1.2, at most two consecutive primes can intersect A in the same prime ideal. If A has Krull dimension r, this means a chain in A can contain no more than r + 1 prime ideals. Thus, the chain P0 ( P1 ( ··· ( Pn can contain at most 2r + 2 ideals. That is, n ≤ 2r + 1. This result is exactly analogous to the bound for general commutative rings. Seidenberg shows in Theorem 2 of [Sei53] that

Theorem 5.3.1. If A is a commutative ring with classical Krull dimension n, then the classical Krull dimension of A[x] is at least n + 1 and at most 2n + 1. 24

As noted, in the centrally closed case, we may assume that every prime intersects A nontrivially. Then a minimal prime must be of the form P [x] for some prime P C A. Thus, the bound is reduced to 2 · k.dim A. We hope that this technique will lead to some sort of inductive proof of a general theorem for Krull dimension of polynomial rings over centrally closed algebras. The problem remains to determine whether it is possible to have a chain of primes in A[x] of the form Pei[x] ( Pi ( Pgi+1[x]. Even more generally, we may ask about the Krull dimension of finite ring extensions. Examples of Stafford in [Sta85] show that the Krull dimension can increase by any amount; however, these examples are only finite over the base ring on one side. Perhaps requiring the extension to be a finite left and right module over the base ring would bound the Krull dimension. Bibliography

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