Abraham Zaks

Atomic rings without a.c.c. on principal ideals

Abstract We study examples of atomic domains which fail to satisfy the a.c.c. on principal ideals.

Annuities under random rates of interest

We investigate the accumulated value of some annuities-certain over a period of years in which the rate of interest is a random variable under some restrictions. We aim at the expected value and the variance of the accumulated value, and we suggest two methods to derive these moments. In some cases both methods have similar difficulties, while in other cases one method is significantly preferable in terms of the simplicity of calculations. The novelty of our second approach lies in the fact that we find recursive relationships for the variance of the accumulated values, and solve these relationships, whilst previous investigators first obtained the second moment.

Geometries of the projective matrix space

In our paper on matrix Mobius transformations (Comm. Algebra 9 (19) (1981), 1913–1968) we introduced the one-dimensional left-projective space over the complex n×n matrices P = P1(Mn(ℂ)). For n = 1 this space is the projective complex line P1(ℂ) and so is homeomorphic to the Riemann sphere. We studied the topology of P and the projective mappings of P onto itself. Here we present some generalizations of certain parts of the Euclidean, the spherical and the non-Euclidean geometry from the scalar to the multidimensional case.

Semiprimary Rings of Generalized Triangular Type

In the study of rings, all of whose residue rings have finite global dimension, S. Eilenberg, H. Nagao, and T. Nakayama have shown in [5] that all residue rings of a semiprimary hereditary ring have finite global dimension. One way to construct semiprimary hereditary rings is to take the ring of triangular matrices over a semisimple ring. This paper is devoted to the study of a class of semiprimary rings that contains all the rings of triangular matrices over semisimple rings. A semiprimary ring R belongs to this class (and we say that R is a T-ring) if each (left) component contains a unique minimal ideal, and each minimal (left) ideal is projective. We start by proving in Section 2 that all residue rings of a T-ring have finite global dimension. In Sections 3 and 4 we investigate the structure of T-rings. In Section 5 we discuss the (left) maximal quotient ring (in the sense of Utumi) of a T-ring, proving that it is a semisimple ring. Further properties of a T-ring R, such as the existence of a simple injective (left) R-module and a characterization of a ring of triangular matrices over a simple ring, are discussed in Section 6. In Section 7 we give some counterexamples concerning the following assertions:

Some rings are hereditary rings

LetR be a bounded Noetherian Prime ring. The Asano-Michler theorem shows thatR is a bounded Dedekind ring if every prime ideal ofR is invertible. We provide a simple proof of the Asano-Michler theorem, and we suggest some possible generalizations. We also prove that if the proper residue rings ofR areQF-rings thenR is a bounded Dedekind ring, and generalize this result toLD-rings.

Reducible sums and splittable sets

Abstract Given the sum of fractions of positive integers s = Σ i = 1 k a i n i , we study conditions under which s is reducible, we provide some examples of irreducible sums, and we investigate the relations between non-splittable sets and irreducible sums.

Geometry of matrix differential systems

In this paper we generalize some basic parts of the projective theory of linear homogeneous differential equations of order m, m > 2, from the scalar to the matrix case. In Section 1 we define the (m l)-dimensional left-projective space over the real n x n matrices P = P, ~ [(M,(R)) and, relying on a previous paper [7] and on standard methods of linear algebra, we state the basic properties of this space. In Section 2 the geometry of the matrix differential systems is described. To each basis of matrix solutions of a given system there corresponds a curve in P and a change of basis corresponds to a projective mapping of this curve. A given curve, together with its projective maps, corresponds to a class of differential systems having projectively equivalent solutions. Some simple properties of these curves are established and a new kind of disconjugacy is defined for the matrix differential systems.

Hereditary Noetherian rings

Abstract Given an Hereditary Noetherian ring, its finitely generated torsion modules are subject to a specified contravariant duality functor, interchanging left and right modules. This duality yields among others many of the well-known results for these rings. For instance, when applied to the ring of integers one recover the structure of the injective torsion abelian groups and their ring of endomorphisms, as well as the structure theory for finitely generated torsion abelian groups. These results follow from the duality. It turns out that the specified duality in this case is just an isomorphism of a finite abelian group with its character group. A similar consequence results for every commutative Dedekind domain. In particular these rings admit a contravariant functor on their finitely generated torsion module, that is an isomorphism on each such module. Some results are derived in more general cases, for instance, for Noetherian rings of injective dimension one.

Contraction of the Matrix Unit Disk

Abstract A Mobius transformation M that is J -equivalent to a contraction, decreases the pseudochordal distance between every pair of points in the unit (matrix) disk. It may keep unchanged the distance between some pairs, and strictly decrease the distance between others.

On the ernbeddings of projective spaces in lines

The projective matrix line P1 (Mn (C)), the higher dimensional projective matrix space Pm - 1 (Mn (C)), m - 2. and inc projective mappings of each space onto itself were considered in [6, 7, 8, 9] Here we embed I he behavior of the projective mappings under this embedding isstudied The chordal and I he spherical distance were previously onlv defined for the matrix line, the embeuding allows us to define these metrics also for the higher dimensional matrix space. We also compare these meirics with the gap metric