Let R be a commutative ring with identity. A subgroup S of GLn(R), where n ≥ 2, is said to be standard if and only if S contains all the q-elementary matrices and all conjugates of those matrices by products of elementary matrices, where q is the ideal in R generated by Xij,xii — Xjj(i ≠ j), for all (xij) ∊ S. It is known that, when n ≧ 3, the standard subgroups of GLn(R) are precisely those normalized by the elementary matrices. To demonstrate how completely this result can break down for n = 2 we prove that GL2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.

If A is a finite abelian group and ZA its integral group ring, consider units u ∊ ZA which have coefficient sum = 1 and are fixed under the involution a —> a-1, a ∊ A. For an odd regular prime p and a p-group A, it is shown that u ≡ 1 mod p if only if u = π(v)v-p, where v is the same kind of unit, and π is the ring endomorphism given by a —> ap, a∊A.

The purpose of this note is to prove that a regular accessible category has a full regular embedding into a set-valued functor category.

A q-analogue of the orthogonality property of the Bessel functions on the zeros is obtained in terms of a q-integral.

Let D be a PID with infinitely many maximal ideals. J. W. Brewer has asked whether some simple ring extension D[α] of D must have nontrivial Picard group. We show that this question has a negative answer.

This paper presents a new type of fractal surfaces called the Takagi surfaces. These are obtained by summing up pyramids of increasing (doubling) frequencies scaled by a geometric ratio b. The fractal dimension (box dimension) of the graph of these functions is shown to be log 8b/log 2.

It is shown that for totally real number fields the Main Conjecture in Iwasawa-Theory for p = 2 proposed by Fédérer implies the 2-primary part of the Birch-Tate-Conjecture in analogy with the case p odd.

Let δ1,δ2 be two derivations of a C*-algebra. We characterize when δ1δ2 is a derivation, a compact, or a weakly compact operator

In this article, we obtain the stochastic ergodic theorem for general additive processes. That is, we prove that there exists , such that whenever α > 0 and A is a measurable set with μ(A) < ∞, where and {U(ij)}} an arbitrary (two dimensional) semigroup of L1 -contractions. This result generalizes the stochastic ergodic theorem (SET) of U. Krengel and the SET of M. A. Akcoglu and L. Sucheston.

Balinski has shown that the graph of a d-polytope is d-connected. In this note we give a new proof of Balinski's theorem.

Nicholson and Watters have recently investigated rings with projective socles and they have shown, among other things, that a ring R has a projective socle if and only if each matrix ring Mn(R), n > 1, has a projective socle. We generalize this result by showing that if S is an excellent extension of R, then the socle of R is projective if and only if the socle of S is projective. Examples of excellent extensions include, as well as matrix rings Mn(R), skew group rings R * G where G is a finite group and the order of G is invertible in R.

We construct a Banach space X which has no equivalent (wLUR) norm but which has no subspace isomorphic to l∞.

It is shown that the concepts of normal radical as introduced by Jaegermann [2] and of invariant radical as introduced by Anderson, Divinsky and Sulinski [1] are equivalent.

In a recent paper D. Borwein [Math. Z. 183(1983), 483- 487] obtained an upper bound for the lp -norms of some generalized Hausdorff matrices, where the sequence (λn) satisfies the condition λn+1 ≦ λn + c, for some positive c. In this paper we obtained the lp -norms of these generalized Hausdorff matrices for which the mass functions are totally monotone.

Let O1 denote the class of groups G such that every group ring of G (over a field) is an Ore domain. Several approaches to the correction of the proof of a result concerning subrings generated by certain O1-groups are given.