Right radicals for right distributively generated near-rings

For right near-rings the left representation has always been considered the natural one. However, Hanna Neumann [6] constructed her right near-rings by writing the reduced free group on the left of the near-ring. In [2] and [8] Neumann's ideas are placed in a more general setting in the sense that right R-groups are used to define radical-like objects in the near-ring R. The right 0-radical ^r J ₀(R) and the right half radical ^r J _½(R) are introduced in [2] where it is shown that for distributively generated (d.g.) near-rings R with a multiplicative identity and satisfying the descending chain condition for left R-subgroups ^r J ₀(R) = J ₂(R), the 2-radical from left representation. In this article we introduce the right 2-radical, ^r J ₂(R) for d.g. near-rings and discuss some of its properties. In particular, we show that for all finite d.g. near-rings with identity J ₂(R) = ^r J ₂(R).     

When is a centralizer near-ring isomorphic to a matrix near-ring?

Necessary conditions are found for a centralizer near-ring M_A(G) to be isomorphic to a matrix near-ring, where G is a finite group which is cyclic as an M_A(G)-module There are centralizer near-rings which are matrix near-rings. A class of such near-rings is exhibited. Examples of centralizer near-rings which are not matrix near-rings are given.     

Kurosh–Amitsur Right Jacobson Radicals of Type-1 and 2 for Right Near-Rings

Near-rings considered are right near-rings. Let ν ∈ {1, 2}. J ^r _ν , the right Jacobson radical of type-ν, was introduced for near-rings by the first and second authors. In this paper properties of these radicals J ^r _ν are studied. It is shown that J ^r _ν is a Kurosh-Amitsur radical (KA-radical) in the variety of all near-rings R in which the constant part R _c of R is an ideal of R. Thus, unlike the left Jacobson radical of type-1 of near-rings, J ^r ₁ is a KA-radical in the class of all zero-symmetric near-rings. J ^r _ν is not s-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings. Received: April 1, 2007. Revised: July 11, 2007.     

Syntactic Rings

If the state set and the input set of an automaton are Ω-groups then near-rings are useful in the study of automata (see [5]). These near-rings, called syntactic near-rings, consist of mappings from the state set Q of the automaton into itself. If, as is often the case, Q bears the structure of a module, then the zerosymmetric part N₀(A) of syntactic near-rings is a commutative ring with identity. If N₀(A) is a syntactic ring then its ideals are useful for determining reachability in automata (see [1] or [2]). In this paper we investigate syntactic rings.     

Modules over matrix near-rings and the ℐ0-radical

This paper clears up some questions concerning type 0 modules over matrix near-rings and the ₀-radical in matrix near-rings. It is shown that, unlike in the type 2 case, type 0 modules over matrix near-rings may arise in several non-isomorphic ways. As a result, we do not always have the same nice relationship between the ₀-radicals of a near-ring and the corresponding matrix near-ring, as we do for the ₂-radical. All near-rings concerned are zero-symmetric with identity element.     

Research in near-ring theory using a digital computer

In this paper we wish to show how the computer has played a valuable role in research in the theory of near-rings. Basically, the author has used the computer to generate examples of near-rings to be applied for meaningful conjectures and counter-examples. All the near-rings of order less than eight are listed in [2]. Since there is only one non-abelian group of order less than eight, it is natural to still be curious what happens when one tries to construct a near-ring from a non-abelian group. The methods used by the author to construct near-rings from groups will be illustrated on the two non-abelian groups of order 8. Specifically, for each non-abelian group of order 8, it was decided to construct all near-rings enjoying one of the following four properties:

1. near-ring with identity:
2. near-rings without two-sided zero;
3. near-rings with no zero divisors;
4. idempotent near-rings; i.e. near-rings for whichx ²=x for allx.

     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

Zerlegungen der Grenzgruppe des einfach-isotropen Raumes Jn

According to Strubecker an affine metric n-space (J_n,g) over a field F of char. 2 (n2) is called simply-isotropic if Radg is one-dimensional; let Fu be the totally isotropic direction w.r. to g. The group B_g of motions of (J_n,g) contains an invariant (2n–1)-dimensional subgroup G, called thelimit group (Grenzgruppe) which maps planes parallel to Fu into parallel ones. In case n=3, n=5, F= Strubecker [12], [13], [14] gave several factorizations G=G_L oG_R of G into a commutative product of subgroups acting 1-transitively (=regularly) on J_n.This note deals with all factorizations of this kind; it followes that n must be odd. A factorization is invariant under B_g iff n=3. Finally it is shown for each factorization of G how the space J_n can be made into a two sided incidence group and into a kinematic space in the sense of Karzel [5], and how the two structures are related to each other.

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Herrn Professor Dr. Karl Strubecker zum 75. Geburtstag gewidmet     

Über die Wiener-Eigenschaft bei projektiven Limites lokalkompakter Gruppen

Leptin posed in [1] the problem to determine the class [W] of locally compact groups G characterized by the following property: Every proper closed two-sided idealJ in the Banach-*-algebraL ¹(G) is annihilated by some nondegenerate continuous *-representation ofL ¹(G) in a Hilbert space. Our main result: A locally compact group G, which is representable as a projective limit of a system of factor groups G/k, k compact normal subgroups, lies in [W] if and only if all the G/k are in [W].     

Radical theory in varieties of near-rings in which the constants form

Not all the good properties of the Kurosh-Amitsur radical theory in the variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of O-symmetric near-rings the theory is much more satisfactory. In this note we show that many of the results of the 0-symmetric near-ring case can be extended to a much bigger variety of near-rings which, amongst others, includes all the O-symmetric as well as the constant near-rings. The varieties we shall consider are varieties of near-rings, called Fuchs varieties, in which the constants form an ideal. The good arithmetic of such varieties makes it possible to derive more explicit conditions.

(i) for the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities),

(ii) for semisimple classes to be hereditary.

We shall prove that the subvariety of 0-symmetric near-rings has attainable identities in a Fuchs variety, and extend the theory of overnilpotent radicals of 0-symmetric near-rings to the largest Fuchs variety F

The near-ring construction of [7] will play a decisive role in our investigations.     

On invisible elements of the Tate-Shafarevich group

Mazur [7] has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the Birch-Swinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J₀ (p) and J₁ (p) that are not visible.     

A representation formula of sum rules for zeros of polynomials

A representation formula in terms of Lucas polynomials of the second kind in several variables (see formula (4.3)), for the sum rulesJ _s ⁽ⁱ⁾ introduced by K.M. Case [1] and studied by J.S. Dehesa et al. [2]–[3] in order to obtain informations about the zeros’ distribution of eigenfunctions of a class of ordinary polynomial differential operator, is derived. Lavoro eseguito nell’ambito del G.N.I.M. del C.N.R.     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

Existence ofBV absolute discontinuous minima for modified multiple integrals of the calculus of variations

We continue here the discussion on the existence of discontinuousBV minima, for a class of multiple integrals of the calculus of variationsI, we have started in [2] in view of possible studies on hyperbolic partial differential equations. Besides the associated Serrin integralJ, based onL ₁-convergence, we take into consideration a modified Serrin-type functionalJ ^*. This new integralJ ^* will be needed in [3] to prove Rankine-Hugoniot type properties.     

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π₂X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T _π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T _π-compact provided G/T _πG is π-complete, extending the abelian group result in a natural way.     

On Prime and Semiprime Near-Rings with Generalized Derivation

Abstract

Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if [F(x),x] = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) on S if [F(x),F(y)] = [x,y] for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.     

Graded Nilpotent Algebras and Eggert's Conjecture

A group G is (l,m,n)-generated if it is a quotient group of the triangle group T(l,m,n) = (x,y,z|x ^l= y ^m= z ⁿ= xyz= 1). In [8] the problem is posed to find all possible (l,m,n)-generations for the non-abelian finite simple groups. In this paper we partially answer this question for the Janko group J ₃. We find all (2, 3, t)-generations as well as (2, 2,2,p)-generations, p a prime, for J ₃     

A construction of Lie algebras by generalized Jordan triple systems of second order

U. Hirzebruch [2] has generalized the Tits' construction of Lie algebras by Jordan algebras [6, also cf. 3, 5] to Jordan triple systems. We show that Hirzebruch's construction of Lie algebras by Jordan triple systems is still valid for generalized Jordan triple systems of second order due to I.L. Kantor [4]. Next, for a given generalized Jordan triple system J of second order, it is shown that the direct sum vector space J⊕J becomes a generalized Jordan triple system of second order with respect to a suitable product, from which we can essentially obtain the same one as the generalization of Hirzebruch's construction.     

The existence of J 3 and its embeddings in E 6

We determine the embeddings of the third sporadic group J ₃ of Janko in simple Chevalley groups of type E ₆ over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J ₃. This is of interest as J ₃ is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].To Jacques Tits on his sixtieth birthdayPartially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.