Robert L. Snider

K0 and noetherian group rings

Abstract The authors use K -theoretic methods to prove that if F is a field of char 0 and G is a torsion free polycyclic-by-finite-group then F [ G ] is a domain.

Rings generated by their units

In 1953 K. Wolfson [14] and in 1954 D. Zelinsky [15] showed, independently, that every element of the ring of all linear transformations of a vector space over a division ring of characteristic not 2 is a sum of two nonsingular ones. Motivated by this result, in 1958, Skornyakov [12, p. 1671 conjectured that all regular rings in which 2 is a unit are generated by their units. The conjecture was finally settled in the negative by G. Bergman [7] in 1974. At about the same time [6], we proved the conjecture in the affirmative for a large class of regular rings; that being, the class of regular rings which are strongly a-regular. Included in this class are those regular rings with primitive factor rings Artinian. In fact, we prove in Theorem 1 that they are unit regular from which the generation by units follows immediately. As a corollary we obtain that regular rings which satisfy a polynomial identity are unit regular. Our best theorem-Theorem 3 -states that if R is strongly n-regular and 2 is a unit in R, then each element of R can he expressed as a sum of two units. The class of stronlgy p-regular rings includes algebraic algebras, locally Artinian rings, and rings with prime factor rings Artinian. Throughout, R will denote a ring with unity element. A ring R is called (van Neumann) regular if for each a in R there exists an x in R such that a = axa. An element a in R is unit regular if there exists a unit u in R such that a = am and R is called unit regular if each element of R is unit regular. In [3] G. Ehrlich proved that the class of unit regular rings includes semisimple Artinian rings, regular rings with no nonzero nilpotent elements, and regular group algebras over fields. The following theorem produces a sufficient condition for regular rings to he unit regular.

SEMIPRIMITIVE π-REGULAR RINGS OF BOUNDED INDEX ARE LEFT MAX RINGS

A ring R is a left max ring if each left R module has a maximal submodule. Left perfect rings are left max rings. In the commutative case, Hamsher has shown that a commutative ring R is a left max ring if and only if the Jacobson radical JðRÞ is left T-nilpotent and R=JðRÞ is von Neumann regular [6]. A ring R is a V ring if each simple module is injective. V rings are left max rings. Noncommutative max rings can look very different. Cozzens has given an example of a noncommutative domain which is a V ring [3]. In any left max ring, Bass has shown that the Jacobson radical of any left max ring is left T-nilpotent [2, p. 470 Remark]. In fact, a ring R is a left max ring if and only if JðRÞ is left T-nilpotent and R=JðRÞ is a left max ring [2, p. 470]. Nevertheless, there have been various noncommutative generalizations of Hamsher’s theorem. Armendariz and Fisher have shown that a left max ring which satisfies a polynomial identity is p-regular [1] and that the converse is true if R has left T-nilpotent Jacobson radical and R=JðRÞ is actually von Neumann regular. This can be extended to show that any von Neumann regular ring with all primitive factors Artin is a V ring and hence a left max ring [5, Proposition 6.18]. More recently Tuganbaev has shown that a ring R which is locally module-finite over its center is a left max ring if and only if JðRÞ is left T-nilpotent and R is strongly p-regular. And Hirano has asked if a p-regular ring satisfying a polynomial identity and having a left

Some Comments on the Jacobson Conjecture

Noetherian rings with Krull dimension one are shown to have closed left ideals in the J-adic topology. The radical of these rings also satisfies the AR property.

Invariant ideals of commutative rings

Let R be a commutative Noetherian ring and G a group of elements acting on R as automorphisms. In this note, we are concerned with the structure of the lattice of invariant ideals of R. In particular we shall compute the Krull dimension of this lattice. Our group is an arbitrary group. There are none of the usual assumptions of some sort of algebraic action. By Krull dimension, we mean the notion of Krull dimension introduced by Rentschler and Gabriel [1, p. 180]. This definition attaches an ordinal number to a Noetherian lattice (and to certain other lattices). (For a definition, see below.) For a commutative Noetherian ring, the Krull dimension of the lattice of ideals is the same as the usual (classical) Krull dimension in terms of the longest chain of prime ideals [1, p. 192]. It follows that in some sense the structure of the lattice of ideals is determined by the poset of prime ideals. The point of this paper is that the same holds for the lattice of invariant ideals. However there may not be enough invariant prime ideals but there are enough prime ideals with finite orbit. We recall that an ideal with finite orbit is called an orbital ideal [2]. The Krull dimension of a lattice can be an infinite ordinal. One can define the classical Krull dimension so that the classical Krull dimension can be an infinite ordinal [1, p. 191]. (For a definition, see below.) We will denote the Krull dimension of the lattice of invariant ideals by KG(R) and the classical Krull dimension of the poset of orbital primes by dG(R). Thus if there is a bound on the lengths of chains of prime ideals, dc{R) is the maximum length of a chain. If there is no bound, then do(R) will be an infinite ordinal.

Periodic groups whose simple modules have finite central endomorphism dimension

Theorem. If k is an uncountable field and G is a periodic group with no elements of order the characteristic of k and if all simple k[G] modules have finite central endomorphism dimension, then G has an abelian subgroup of finite index.

Endomorphism rings of simple modules over group rings

If N is a finitely generated nilpotent group which is not abelianby-finite, k a field, and D a finite dimensional separable division algebra over k, then there exists a simple module M for the group ring k[G] with endomorphism ring D. An example is given to show that this cannot be extended to polycyclic groups. Let N be a finite nilpotent group and k a field. The Schur index of every irreducible representation ofN is at most two. This means that for every irreducible module M of the group ring k[N ], Endk[N ](M) is either a field or the quaternion algebra over a finite field extension of k [6, p.564]. The purpose of this paper is to examine the situation when N is an infinite finitely generated nilpotent group. In this case, the endomorphism rings of simple modules are still finite dimensional [5, p.337]. However the situation is much different and in fact we prove Theorem 1. Let k be a field, N a finitely generated nilpotent group which is not abelian-by-finite, and D a separable division algebra finite dimensional over k, then there exists a simple k[N ]-module M with Endk[N ](M) = D. I do not know if D in Theorem 1 can be any inseparable division algebra. The calculations seem quite difficult for large inseparable division algebras although the calculations below would show that small inseparable algebras (where the center is a simple extension) do occur. On the other hand, if N is abelian-by-finite, then the group ring k[N ] satisfies a polynomial identity of some degree n. Since endomorphism rings of simple modules are homomorphic images of subrings of k[N ], each endomorphism ring of a simple module must satisfy the same identity of degree n and hence must be a division ring of degree at most n/2. If G is a polycyclic group which is not nilpotent-by-finite, one expects that somehow G will be “more noncommutative” and hence that it should be easier to get noncommutative division rings as endomorphism rings. However we will give an example of a polycyclic group G, not nilpotent-by-finite, and a field k such that there are finite dimensional separable division algebras over k which are not the endomorphism ring of any simple k[G]-module. (The endomorphism rings of simple modules are still finite dimensional [5, p. 337].) Received by the editors October 17, 1994. 1991 Mathematics Subject Classification. Primary 16S34, 20C05; Secondary 16K20, 16S50.