Rationality for generic toric rings

We study generic toric rings. We prove that they are Golod rings, so the Poincaré series of the residue field is rational. We classify when such a ring is Koszul, and compute its rate. Also resolutions related to the initial ideal of the toric ideal with respect to reverse lexicographic order are described. Received August 13, 1997; in final form October 23, 1998     

The centers of generic division algebras with involution

The main goal of this paper is a study of the centers of the generic central simple algebras with involution. These centers are shown to be invariant fields under finite groups in a way analagous to the center of the generic division algebras. The centers of the generic central simple algebras with involution are also described as generic splitting fields (i.e. function fields of Brauer-Severi varieties) over the centers of generic division algebras. Finally, a generic central simple algebra is described for the class of central simple algebras with subfields of a certain dimension. The first author would like to thank the Department of Mathematics of The University of Texas at Austin for its hospitality and the NSF for its support under grant DMS 585-05767. The second author would like to thank the NSF for its support under grants DMS 8303356 and DMS 8601279.     

Dimensions of division rings

Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim _k g are invariants for the respective division rings, by calculating the Krull and global dimensions ofD(G)? _k D(G) andD(g)? _k D(g).     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

A remark on generic splitting rings

Archiv der Mathematik -     

On embedding of group rings in division rings

LetF be a free group,N⊲F andV(N) be a verbal subgroup ofN. For the group ringR , whereR is any field and F/V(N), the zero divisor problem of Kaplansky and the problem of embeddingR in a division ring are investigated. It is proved, in particular, thatR has no zero divisors and can be embedded in a division ring whenF/N is finitely approximated andN/V(N) is approximated by nilpotent groups without torsion.     

Homological properties of generic matrix rings

It is shown that the ring of two 2×2 generic matrices over a field has infinite global dimension. It is also proved that there is a non-free projective module over that ring. Finally, the authors show that the trace ring of that generic matrix ring is an iterated Ore extension from which it follows that the trace ring has global dimension five and that the finitely-generated projective modules are stably free.     

Finite-dimensional subalgebras of division rings

For any division ringD and any two simple Artinian algebras finite dimensional overF=Center(D) we characterize the minimal size of anF-extension ofD that contains commuting images of these algebras. In particular we show that ifD contains subalgebras of coprime dimensionsn andm then they have commuting conjugates inD, andD contains a subalgebra of dimensionnm.     

Multiplicative commutators in division rings

In this paper we study division rings in which the multiplicative commutators are periodic or periodic relative to the center. This work was supported by NSF grant MCS-76-06683 at the University of Chicago. This paper was written while the author was a vistor at the Institute for Advanced Studies, The Hebrew University of Jerusalem.     

Strongly regular rings and rational identities of division rings

We look at division rings in the variety of strongly regular rings and show a connection to the study of rational identities on division rings.Research partially supported by a Grant from NSERC.     

A dimension theorem for division rings

LetD be a division algebra over a fieldk, letn be an arbitrary positive integer, and letk(x ₁,...,x _n) denote the rational function field inn variables overk. In this note we complete previous work by proving that the following three conditions are equivalent: (i) there exists an integerj such that the matrix ringM _j(D) contains a commutative subfield which has transcendence degreen overk; (ii) K dim (D⊗_k k(x ₁,...,x _n )) =n; (iii) gl. dim (D⊗_k k(x ₁,...,x _n )) =n. The crucial tool in the proof of this theorem is the Nullstellensatz forD[x ₁,...,x _n] which was obtained by Amitsur and Small.     

Multiplicative commutators in division rings II

SiaD un corpo,Z il centro diD, e supponiamo cheZ non sia numerabile. Allora:

1. Se tutti i commutatorix y x ^?1 y^?1 sono periodici rispetto aZ, D risulta un campo.
2. SeN è un sottogruppo sottonormale diD che è periodico rispetto aZ,N risulta centrale.

Questi risultati ci portano più vicino alla soluzione di congetture fatte da noi in un articolo precedente.     

Finite groups embeddable in division rings

In a tour de force in 1955, S. A. Amitsur classified all finite groups that are embeddable in division rings. In particular, he disproved a conjecture of Herstein which stated that odd-order emdeddable groups were cyclic. The smallest counterexample turned out to be a group of order 63. In this note, we prove a non-embedding result for a class of metacyclic groups, and present an alternative approach to a part of Amitsur's results, with an eye to ``de-mystifying" the order 63 counterexample.