D. S. Passman

Units in integral group rings

Abstract Let V = V ( Z [ G ]) denote the group of normalized units in the integral group ring Z [ G ] of the finite group G . In this paper, we show that G has a torsion-free normal complement N in V provided G is either the circle group of a nilpotent ring or that G has an abelian subgroup of index at most 2. The main difficulty is to prove in the latter case that N is torsion-free.

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringRG are equal provided •G•−∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.

Noetherian down-up algebras

Down-up algebras A = A(α, β, γ) were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that β 6= 0 is equivalent to A being right (or left) Noetherian, and also to A being a domain. Furthermore, when this occurs, we show that A is Auslander-regular and has global dimension 3.

Crossed products over prime rings

In this paper we obtain necessary and sufficient conditions for the crossed productR *G to be prime or semiprime under the assumption thatR is prime. The main techniques used are the Δ-methods which reduce these questions to the finite normal subgroups ofG and a study of theX-inner automorphisms ofR which enables us to handle these finite groups. In particular we show thatR *G is semiprime ifR has characteristic 0. Furthermore, ifR has characteristicp>0, thenR *G is semiprime if and only ifR *P is semiprime for all elementary abelianp-subgroupsP of Δ+(G) ∩Ginn.

A Course in Ring Theory

Projective modules: Modules and homomorphisms Projective modules Completely reducible modules Wedderbum rings Artinian rings Hereditary rings Dedekind domains Projective dimension Tensor products Local rings Polynomial rings: Skew polynomial rings Grothendieck groups Graded rings and modules Induced modules Syzygy theorem Patching theorem Serre conjecture Big projectives Generic flatness Nullstellensatz Injective modules: Injective modules Injective dimension Essential extensions Maximal ring of quotients Classical ring of quotients Goldie rings Uniform dimension Uniform injective modules Reduced rank Index.

Group algebras whose units satisfy a group identity. II

Let K[G] be the group algebra of a torsion group G over an infinite field K, and let U = U(G) denote its group of units. A recent paper of A. Giambruno, S. K. Sehgal, and A. Valenti proved that if U satisfies a group identity, then K[G] satisfies a polynomial identity, thereby confirming a conjecture of Brian Hartley. Here we add a footnote to their result by showing that the commutator subgroup G′ of G must have bounded period. Indeed, this additional fact enables us to obtain necessary and sufficient conditions for U(G) to satisfy an identity.

Computing the symmetric ring of quotients

Abstract We discuss and compute the symmetric Martindale ring of quotients for various classes of prime rings. In particular, we consider free algebras and group algebras.

Centers and prime ideals in group algebras of polycyclic-by-finite groups

Let G be a polycyclic-by-finite group and let K be a field. Then a well-known theorem of P. Hall [7, Corollary 10.24 asserts that the group algebra K[q is both a right and left Noetherian ring. In particular, if P is a prime ideal of K[C;I, then K[C;I/P is a prime Noetherian ring and work of Goldie [3, Theorem 1.371 implies that K[C;I/P has a classical right ring of quotients Z?(K[q/P) which is simple Artinian. In analogy with [I], we define the heart of P, S(P), to be the the center of this ring of quotients. Thus S(P) is a field containing K and the aim of this paper is to study the extension &‘(P)/K. We show first that if P is a particularly nice prime, a standard prime, then S(P) is equal to 2?(9’(K[q/P)), the ring of quotients of the center of K[C;I/P. Furthermore, the latter is a finite extension of the image of the center of K[Gj in K[G]/P. Now it follows from a recent important paper of Roseblade [8] that every prime P is closely related to a standard prime. Because of this, we can therefore show that for all such P, S(P) is a finitely generated extension of K. In particular, the transcendence degree of Z(P) over K is finite and we denote this number by c.r. P, the central rank of P. We also show that if N u G and if P is a prime ideal of K[G], with H(P) a nonabsolute field, then there exists a prime Q of K[N] with c.r. Q < c.r. P and P n K[NJ = flrEC p. Since the primitive ideals of K[G] are essentially those primes of central rank 0, we therefore obtain a number of corollaries concerning the latter ideals.

Radicals of crossed products of enveloping algebras

LetL be a Lie algebra over a fieldK which acts asK-derivations on aK-algebraR. Then this action determines a crossed productR *U(L) whereU(L) is the enveloping algebra ofL. The goal of this paper is to describe the Jacobson radical ofR * U(L) forL≠0. We are most successful whenR is a p.i. algebra or Noetherian. In more general situations we at least obtain upper and lower bounds forJ(R * U(L)) which are ideals extended fromR. Furthermore, we offer an interesting example in all characteristics of a commutativeK-algebraC which admits a derivationδ such thatC isδ-prime but not semiprime.

Enveloping algebras satisfying a polynomial identity

In this paper we study ordinary and restricted enveloping algebras satisfying a polynomial identity. We first show how the Δ-methods of J. Bergen and D. S. Passman (J. Algebra, in press) can be used to handle U(L) in all characteristics and u(L) if the latter ring is prime. In particular, we offer a simpler proof of the result of Yu. A. Bachturin (J. Austral. Math. Soc. 18, 1974, 10–21) on ordinary enveloping algebras in characteristic p > 0. Our main theorem asserts that, in general, u(L) is p.i. if and only if L has a restricted subalgebra of finite codimension which is (essentially) commutative. These results are clearly the Lie algebra analogs of the p.i. group ring theorems of M. K. Smith (J. Algebra 18, 1971, 477–499) and D. S. Passman (Pacific J. Math. 36, 1971, 467–483).