MODULO-CONSTANT IDEAL-HEREDITARY RADICALS OF NEAR-KINGS

ABSTRACT

It is known that no “good” radical of (not necessarily o-symmetric) near-rings can be ideal-hereditary. Using the results of the o-symmetric case, we show that the situation is not as bad as on first appearances and we give several examples of (Kurosh-Amitsur) radicals of near-rings for which the semisimple class is hereditary and the radical class is hereditary on left invariant ideals. We also extend some recent results on left strong radicals from the o-symmetric case to the general case.     

On a Question of J. H. Meyer

Abstract

In this paper, we present a partial solution to the following question of J. H. Meyer (Meyer, J. H. (1986). Matrix Near-Rings. Ph.D. dissertation, University of Stellenbosch, South Africa, Prob. 11): Find a necessary and sufficient condition for a near-ring to be isomorphic to a matrix near-ring. In fact, three characterization results for abstract affine matrix near-rings are given. As a corollary, we get that, for each n ≥ 2, the class of all n × n matrix near-rings over abstract affine near-rings is finitely axiomatizable.     

STRONGLY EQUIPRIME NEAR-RINGS

Abstract

Strongly equiprime near-rings are defined which generalize strongly prime rings to near-rings. These near-rings determine an ideal-hereditary Kurosh-Amitsur radical in the variety of 0-symmetric near-rings. In the same variety, the uniformly strongly equiprime near-rings also determine an ideal-hereditary Kurosh-Amitsur radical which is not comparable with the Jacobson-type radicals nor with the Brown-McCoy-type radicals.     

SPECIAL RADICALS OF NEAR-RING MODULES

Abstract

Equiprime near-rings, which generalize the concept of prime-ness in rings, were defined by the present authors, together with S. Veldsman. This concept was shown in subsequent work to lead to a very satisfactory theory of special radicals for near-rings. In the current paper, we define equiprime N-groups for a near-ring N. It is shown that an ideal A of N is equiprime if and only if it is the annihilator of an equiprime TV-group G. Special classes of near-ring modules are defined, and a module-theoretic characterization of special radicals of near-rings is established, similar to that given by Andrunakievich and Rjabuhin for special radicals of rings.     

PIECEWISE ENDOMORPHISMS OF RING MODULES

Abstract

We study centralizer near-rings of ring modules which are rather special in two respects. Firstly, the elements of the near-rings are piecewise endomorphisms of the modules concerned. and secondly, the near-rings themselves are, in fact, rings.     

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.     

MORITA NEAR-RINGS

Abstract

We define a morita context for near-rings and subsequently a morita near-ring. Any near-ring can be considered as a morita context for near-rings, and the corresponding morita near-ring is just the 2 x 2 matrix near-ring over the near-ring. We also determine the relationships between the ideals of a morita context and the ideals of a morita near-ring.     

Distributively generated gc near-rings

Let N be a finite GC near-ring. Those near-rings N such that N is distributilvely generated and (N,+) is solvable are shown to be a direct sum of fields and d.g. ” basic near-rings of size 2“. These basic near-rings of size 2 are characterized. A method for constructing d.g. GC near-rings is presented. This work gives rise to a class of d.g. GC near-rings which are not centralizer near-rings.     

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.

     

EQUIPRIME Γ-NEAR-RINGS

Abstract

Equiprime and strongly equiprime near-rings were recently defined by the present authors, together with S. Veldsman. In the present paper, the concepts are introduced for Γ-near-rings, and give rise to Kurosh- Amitsur radicals. If M is a Γ-near-ring and L is its left operator near-ring, then R(L)⁺ = R(M), where R(—) in both cases denotes either the equiprime or the strongly equiprime radical.     

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.     

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.     

ON NEAR-RINGS WITH MATRIX UNITS

Abstract

In ring theory it is well known that a ring R with identity is isomorphic to a matrix ring if and only if R has a set of matrix units. In this paper, the above result is extended to matrix near-rings and it is proved that a near-ring R with identity is isomorphic to a matrix near-ring if and only if R has a set of matrix units and satisfies two other conditions. As a consequence of this result several examples of matrix near-rings are given and for a finite group (Γ, +) with o(Γ) > 2 it is proved that M₀ (Γⁿ) is (isomorphic to) a matrix near-ring.     

Distributor and j2-radical ideals of generalized centralizer near-rings

In 1980, Maxson and Smith [1] determined the J₂-radical ideal for the ceiitralizer near-ring M_A(G), where A is a group of automorphisms over a group G. Further, in 1985, Smith [4] generalized MA(G) to the class of generalized ceiitralizer near-rings. In this paper we determine both the J₂-radical and the distributor ideals for the class of generalized ceiitralizer near-rings. We further push these results to determine all the homomorphie images of generalized ceiitralizer near-rings.     

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.     

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).     

SUBDIRECT IRREDUCIBILITY AND RADICALS

Abstract

We prove lemmas in Andrunakievich s-varieties on the transitivity of the relation “is an ideal of” and concerning subdirectly irreducible factor rings. Applying these lemmas we show that a Plotkin radical introduced in [8] has the ADS-property and is ideal hereditary. These lemmas are applicable in proving a subdirect decomposition for rings having an ideal with 0 antisimple radical. For Jordan algebras and near-rings (they do not form Andrunakievich varieties) we can prove a similar subdirect decomposition concerning ideals with 0 Brown-McCoy radical.     

Minimal left ideals of near-rings

We show that a finite minimal left ideal L of a zero symmetric near-ring N is a planar near-ring if L is not contained in the radical J ₂(N). This result will follow from a more general discussion on minimal N-subgroups of a near-ring. Then we discuss some consequences of this result when applied to the structure theory of near-rings. Finally we transfer our results to rings and deal with some ring theoretic questions concerning “trivial” multiplications in rings.     

On the summands of near-rings with ATM

We say that a near-ring (N,+,·) has an almost trivial multiplication (ATM) if the product of two elements belongs to the intersection of the additive cyclic groups generated by these two elements. We show that every finite near-ring with ATM can be decomposed to a direct sum where the summands are either near-rings defined on cyclic groups or near-rings whose minimal ideals are zero near-rings. Finally, we show how to construct these summands on cyclic groups.     

On equiprime near-rings

An equiprime near-ring is a generalization of prime ring. Firstly some axioma-tics concerning equiprime near-rings are discussed, e.g. their relation to the other notions of primeness for near-rings, primitive near-rings and near-fields. Secondly we investigate the equiprimeness of some well-known examples of near-rings.