# Publications

1. Goldie dimension of prime factors of polynomial and skew polynomial rings, J. London Math. Soc. (2) 29 (1984) 418-424. (MR 85k:16027)
Abstract (in LaTeX)

We show that if $R$ is a right Noetherian ring containing the rational numbers, any bound on the Goldie dimension of prime factors of $R[x]$ is actually a bound on the Goldie dimension of prime factors of any polynomial ring $R[x_1,\dots,x_m]$. A result of Sigurdsson then shows that the same bound holds for any iterated differential operator ring $R[\theta;\delta_1]\dots[\theta;\delta_m]$.

2. When are all prime ideals in an Ore extension Goldie?, Comm. Algebra 13 (1985) 1743-1762. (MR 86j:16003)
Abstract (in LaTeX)

We show that if $\phi$ is an automorphism of $R$, then all prime ideals of the skew polynomial ring $R[x;\phi]$ are right Goldie if and only if (a) every prime ideal of $R$ is right Goldie and (b) every strongly $\phi$-prime ideal of $R$ is a finite intersection of prime ideals. [We say $I$ is strongly $\phi$-prime if $\phi(I)=I$ and if whenever $\phi^n(J)K$ is contained in $I$ for ideals $J,K$ and all large $n$, we have either $J$ or $K$ contained in $I$.] We obtain a similar result for skew Laurent rings, and we obtain some partial results for skew polynomial rings with both automorphism and derivation.

We show that if $R$ has the a.c.c. on ideals, then all prime ideals in $R[x;\phi]$ are right Goldie if and only if all prime ideals in $R$ are right Goldie, but we give an example showing that this is not true in general. We give some other negative examples.

3. Localization and ideal theory in Noetherian strongly group-graded rings, J. Algebra 105 (1987) 76-115. (MR 88c:16105)
Abstract (in LaTeX)

We study primality, hypercentrality, simplicity, and localization and the second layer condition in skew group rings and group-graded rings. We give necessary and sufficient conditions for the skew group ring of a torsion-free nilpotent group to be a simple ring, and if the coefficient ring is commutative, we give necessary and sufficient conditions for the skew group ring of an Abelian group to be simple. Our method involves showing certain group-graded rings are hypercentral. Our main results show that if $G$ is a polycyclic-by-finite group and $R$ is an Artinian ring or a commutative Noetherian ring, then a strongly $G$-graded ring with base ring $R$ satisfies the second layer condition. We discuss consequences of this for localization in such rings.

4. Localization and ideal theory in iterated differential operator rings, J. Algebra 106 (1987) 376-402. (MR 88m:16002)
Abstract (in LaTeX)

We study primality, hypercentrality, simplicity, and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings. We give sufficient conditions for the skew enveloping algebra of a nilpotent Lie algebra with coefficient ring containing the rational numbers to be a simple ring, and we give necessary and sufficient conditions in the case that the Lie algebra is Abelian. Our main results show that if $L$ is a finite dimensional solvable Lie algebra and $R$ is an Artinian ring or a commutative Noetherian algebra over $k$, then the skew enveloping algebra $R\#U(L)$ satisfies the second layer condition. We discuss consequences of this for localization and use the localization theory to state a classical Krull dimension versus global dimension inequality when $k$ is uncountable.

5. (with K. R. Goodearl) Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions, Pacific J. Math. 131 (1988) 13-37. (MR 88j:16004)
Abstract (in LaTeX)

This paper is principally concerned with the question of whether a generalized differential operator ring $T$ over a ring $R$ must have the same uniform rank (Goldie dimension) or reduced rank as $R$, and with the corresponding questions for induced modules. In particular, when $R$ is either a right or left noetherian $\bbQ$-algebra, or a right noetherian right fully bounded $\bbQ$-algebra, it is proved that $T_T$ and $R_R$ have the same uniform rank. For any right noetherian ring $R$, it is proved that $T_T$ and $R_R$ have the same reduced rank. The type of generalized differential operator ring considered is any ring extension $T\supseteq R$ generated by a finite set of elements satisfying a suitable version of the Poincarè-Birkhoff-Witt Theorem.

6. Notes on Localization in Noncommutative Noetherian Rings, Cuadernos de Algebra 9, Univ. of Granada, 1989, 77 pp.

This is a monograph that is meant to be a primer on localization in noncommutative noetherian rings, with emphasis on the second layer condition of Jategaonkar and on localizing at prime ideals and at cliques of prime ideals. Various technicalities relating to this material are presented, along with a large bibliography.

7. (with G. Sigurdsson) Catenarity and Gelfand-Kirillov dimension in Ore extensions, J. Algebra 127 (1989) 409-425. (MR 91b:16027)
Abstract (in LaTeX)

If $R$ is a commutative affine domain (for example, the coordinate ring of an irreducible algebraic variety), the sum of the height of any prime ideal in $R$ and the dimension of the corresponding factor ring is the dimension of $R$. This implies that if $Q$ and $P$ are prime ideals of $R$ with $Q\supseteq P$, any saturated chain of prime ideals from $P$ to $Q$ has length $\dim R/P – \dim R/Q$, and so $R$ is catenary. Schelter, Gabber, and others have proven analogous statements for some noncommutative affine rings $R$. In this paper we study the question of when an Ore extension of a commutative ring is catenary, giving both positive and negative results. We show that for locally finite dimensional Ore extensions of commutative affine rings, the above dimension statements are still true if we use the Gelfand-Kirillov dimension.

8. (with I. M. Musson) Primitive factors of enveloping algebras of nilpotent Lie superalgebras, J. London Math. Soc. (2) 42 (1990) 401-408. (MR 92b:17013)
Abstract (in LaTeX)

We show that if $U$ is the enveloping algebra of a finite-dimensional nilpotent Lie superalgebra over a field of characteristic zero, then any graded-primitive factor ring of $U$ is isomorphic to a tensor product $C\otimes_k A$ where $C$ is the Clifford algebra of a nonsingular form over some finite field extension of $k$ and $A$ is a Weyl algebra over $k$. We prove that the same result holds for a primitive factor of $U$, except that $C$ may be either the whole Clifford algebra or just its even part. We give examples to show all possibilities can occur. Our results generalize a result of Dixmier for ordinary Lie algebras.

9. A criterion for primeness of enveloping algebras of Lie superalgebras, J. Pure and Applied Algebra 69 (1990) 111–120. (MR 92b:17014)
Abstract (in LaTeX)

We show that if the determinant of a certain matrix obtained from the Lie product on the odd part of a Lie superalgebra is nonzero, then the enveloping algebra of the Lie superalgebra is a prime ring. We then apply this criterion to show that the enveloping algebra of a classical simple Lie superalgebra not of type $b(n)$ is a prime ring.

10. Skew differential operators on commutative rings, in Abelian Groups and Noncommutative Rings: A collection of papers in memory of Robert B. Warfield, Jr., ed. L. Fuchs, K. R. Goodearl, J. T. Stafford, and C. Vinsonhaler, Contemporary Mathematics 130, American Mathematical Society, Providence, 1992, pp. 49-67. (MR 93h:13024)
Abstract (in LaTeX) PDF preprint

We define a ring of skew differential operators on a commutative ring $A$ using a commutator twisted via the powers of an automorphism $\phi$ of $A$, derive some of the basic properties of this construction, and work out some examples. We show that many of the standard properties of differential operators continue to hold with our definition but that a crucial difference occurs for automorphisms of infinite order: frequently the action of a skew differential operator is then determined by its action on the powers of a single element of $A$.

11. (with R. Farnsteiner) An application of Frobenius extensions to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993) 407-424. (MR 93c:17049)
Abstract (in LaTeX)

By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory.

12. (with K. R. Goodearl) Algebras of bounded finite dimensional representation type, Glasgow Math. J. 37 (1995) 289-302. (MR 97c:16019)
Abstract (in LaTeX) PDF preprint

It is shown that for an arbitrary affine or noetherian algebra $R$ over a field, bounded representation type for the finite dimensional $R$-modules implies finite representation type for such modules. In fact, this boundedness assumption guarantees the existence of an ideal $I$ annihilating all finite dimensional
$R$-modules such that $R/I$ is a finite dimensional algebra of finite representation type. Bounds on lengths of certain classes of finite-length modules are also investigated. For example, if $R$ is a noetherian ring satisfying the second layer condition and admitting a finite bound on the lengths of the indecomposable finite-length $R$-modules having co-artinian annihilators, it is proved that $R$ is a direct product of an artinian ring of finite representation type and a ring with no proper co-artinian ideals.

13. (with S. S. Stalder and M. L. Teply) Prime ideals and radicals in semigroup-graded rings, Proc. Edinburgh Math. Soc. 39 (1996) 1–25. (MR 97a:16081)
Abstract (in LaTeX) PDF preprint

In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings.

We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup $\Omega$ with preorder defined by $\alpha\lt\beta$ if $\beta$ is in the ideal generated by $\alpha$, and we use the direct sum to construct an $\Omega$\!-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in $\Omega$ when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring $S[\Omega]$ correspond bijectively to pairs $(\Phi,Q)$ with $\Phi$ a prime ideal of $\Omega$ and $Q$ a prime ideal of $S$. The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when $\Omega$ is an infinite free band, by showing that $S[\Omega]$ is prime whenever $S$ is.

When $\Omega$ satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of $\Omega$, and also to the direct product of the component rings.

14. Prime ideals and radicals in rings graded by Clifford semigroups, Comm. Algebra 25#5 (1997) 1595-1608.
(MR 98e:16022)
Abstract (in LaTeX) PDF preprint

In this paper we continue our study of the ideal structure of the direct sum of a directed system of rings indexed by a semigroup begun in “Prime Ideals and Radicals in Semigroup-graded Rings” (with S. Stalder and M. Teply), with emphasis on describing the prime ideals and radicals of semigroup rings and semigroup-graded rings. This time we concentrate on semigroups that fail to satisfy condition $(\dag)$ of our orginal article but have a sufficient quantity of nearly central idempotents, and we reduce the description of the prime ideals and radicals to the case of group rings and prime families over systems of group rings.
Our results apply to Clifford semigroups, commutative semigroups for which every element has a power lying in a subgroup, and some more general classes of semigroups.

15. Comodule algebras and Galois extensions relative to polynomial algebras, free algebras, and enveloping algebras, Comm. Algebra 28#1 (2000) 37-62. (MR 2000m:16047)
Abstract (in LaTeX) PDF preprint

In this paper we study the question of when an $H$-comodule algebra is a faithfully flat Galois extension of its subalgebra of coinvariants for certain Hopf algebras $H$. We note that if $H$ is connected, a faithfully flat Galois extension must actually be cleft and hence a crossed product, and we show that with a different hypothesis, a faithfully flat Galois extension must be a smash product. We also describe faithfully flat Galois extensions when $H$ is pointed cococummutative. We give an explicit description of $H$-comodule algebras when $H$ is a polynomial algebra, a divided power Hopf algebra, a free algebra, or a shuffle algebra. We give necessary and sufficient conditions for an $H$-extension to be faithfully flat Galois in these cases and in the case where $H$ is the
enveloping algebra of a Lie algebra; a key ingredient in our analysis is the existence and description of a total integral. In the case where $H=k[x]$, we give a simple example of a flat Galois extension that is not faithfully flat.