Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

In [8] the author studied the question of the primitivity of an Ore extension R[x, δ], where δ is a derivation of the ring R. If a is an automorphism of R then it can be shown that R[x, α] is primitive if the following conditions are satisfied: (i) no power αsS ≥ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are also known to be necessary and sufficient for the skew Laurent polynomial ring R[x, x−1, α] to be simple [9]. The object of this paper is to find conditions which are sufficient for R[x, x−1, α] to be primitive. The results obtained are remarkably similar to those of [8]. Two logically independent conditions are each found to be sufficient for the primitivity of R[x, x−1, α]. Of these, one is also shown to be sufficient for R[x, α] to be primitive. Included in the examples illustrating these results are some applications to the theory of primitive group rings. The basic techniques involved are also applied to produce a counterexample to the converse of a theorem of Goldie and Michler [3] on when R[x, x−1, α] is a Jacobson ring.

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).

In [1] J. Lindenstrauss and M. A. Perles studied the extreme points of the set of all linear operators T of norm ≤ 1 from a finite dimensional Banach space X into itself. In particular they studied the question “When do these extreme points form a semigroup?”.

Interest in the ranges of Lyapunov transformations began with Taussky in [5]. Recently in a series of papers, Loewy has studied the ranges of Lyapunov transformations on matrices. In particular in [2] and [3], the following result was obtained.

The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.

A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following

Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.

It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].

If R is an integral domain with maximum condition for principal right ideals—right ACC1—every nonzero non-unit in Rhas irreducible factors, but is not necessarily a product of such factors. Using additional basic factors—called infinite primes in [1]—results about unique factorization in principal right ideal domains have been obtained in [1], [2], and [5].

It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the form

where Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact that

is a modular form for a certain congruence subgroup of the modular group, and

with

where Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

In this paper we study several generalizations of the concept of unique factorization domain. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. Theorem 1 shows that the class of π-domains forms a rather natural subclass of the class of Krull domains. In Section 3 we consider overrings of π-domains. In Section 4 generalized GCD-domains are introduced: these form an interesting class of domains containing all Prüfer domains and all π-domains.

to the paper

On a Dual Integral Equation with a Trigonometric Kernel

by D. C. Stocks

Professor W. E. Williams (Surrey University) has recently pointed out to me that my paper is incorrect.

The step from equation (5) to equation (6) implies that φ is independent of λ, which the solution (11) shows is not the case. In view of this, equation (12) is not a solution to the original equations unless

with φ defined by (11).