Showing papers in "Acta Mathematica Hungarica in 1992"

Further results on reticulated rings

Abstract: For more details about the reticulation LR of a ring, see Simmons [9]. For any ideal I of the ring R, let D(I) be the ideal generated by {D(a) : a E I ) in LR. For any ideal J of the lattice LR, let D-l (J ) = {a E R: D(a) E J). Trivially, D-I (J ) is an ideal of R. The reticulation of a ring was investigated by Simmons [9] in order to show that a lo t of ring theoretic properties have analogues in lattice theory and vice versa. In this paper we continue this theme and answer a couple of questions raised by Simmons [9]. Then we proceed to prove further results in that direction. Let Id(R) be the lattice of all ideals of the ring R. It should be noted that this lattice is not necessarily distributive. Let RId(R) be the distributive lattice of radical ideals of the ring R. For a lattice L, let Id(L) be the lattice of ideals of L. From now on, LR will denote the reticulation of R. We start by quoting a theorem that was given by Johnstone [7], p. 194.

Rings with left self distributive multiplication

Abstract: Throughout this note all rings are associative, but not necessarily commutat ive or with unity. A ring is left self distributive (an LD-ring) if it satisfies the identity: x y z = xyx z . Similarly one defines a right self distributive ring (an RD-ring). Petrich [7] classified all rings which are both L D and RD-rings as those rings which are the direct sum of a Boolean ring and a nilpotent ring of index at most three. As shown by several examples given herein, this does not hold for all LD-rings. These examples illustrate how rich the variety of LD-rings is. If R is an LD-ring and N is the set of nilpotent elements of R, then N is an ideal, N 3 = 0, and R / N is Boolean. If R / N contains unity, then R = A q~ N as a direct sum of left ideals, with A a Boolean ring with unity. This condition is implied by several others; e.g., R has d.c.c, on ideals or a.c.c, on ideals. Without any finiteness condition we are still able to find an ideal B = A + N, where A is Boolean and is a left ideal, such that B is completely semiprlme and is left and right essential in R. Somewhat surprising from the viewpoint of semigroup theory [5] is the result that every LD-ring is left permutable (satisfies abc= bac identically). Other useful identities are developed. A complete classification of subdirectly irreducible LD-rings is given. Such a ring is either nilpotent of index at most three, Z2, or a certain four element ring.

Ergodic properties of some inverse polynomial series expansions of laurent series

Abstract: Recently A. Knopfmacher and the present author [8] introduced and studied some properties of various unique expansions of formal Laurent series over a field F, as the sums of reciprocals of polynomials, involving "digits" al, as , . . , lying in a polynomial ring F[X] over F. In particular, one of these expansions (described below) turned out to be analogous to the so-called Liiroth expansion of a real number, discussed in Perron [13] Chapter 4. In a partly parallel way, Artin [1] and Magnus [9, 10] had earlier studied a Laurent series analogue of simple continued fractions of real numbers, involving "digits" Xl,Z2,. . . in a polynomial ring as above. In addition to sketching elementary properties of an n-dimensional "Jacobi-Perron" variant of this, Paysant-Leroux and Dubois [11, 12] also briefly outlined certain "metric" theorems analogous to some of Khintchine [7] for real continued fractions, in the case when F is a finite field. The main aim of this paper is to derive some similar metric or ergodic results for the Laurent series Liirothtype expansion referred to above. (For analogous results concerning Liiroth expansions of real numbers, see :lager and de Vroedt [5] and Salgt [14], and also [16, 17, 18].) In order to explain the conclusions, we first fix some notation and describe the inverse-polynomial Liiroth-type representation to be considered: