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 near-rings with ATM

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 give examples of such near-rings and we show that the prime radical has a strong influence on the structure of the additive group of a near-ring with ATM.     

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.     

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.

     

Additive Groups of Local Near-Rings

The object of this note is to obtain information about the additive groups of local near-rings. It will be shown that the additive group of a torsion local near-ring is a bounded p-group. The torsion Abelian groups which are additive groups of local near-rings will be described completely. A method will be given to construct groups of almost any prime power order, which are not additive groups of local near-rings.     

Medial near-rings

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a Binomial Theorem, show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. Natural examples from analysis and geometry are produced via a general construction method.     

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.     

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.     

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.     

Prime Ideals in Near-Rings

An ideal I of a near-ring R is a type one prime ideal if whenever a Rb ? I, then a ∈ I or b ∈ I. This paper considers the interconnections between prime ideals and type one prime ideals in near-rings. It also develops properties of type one prime ideals, gives several examples illustrating where prime and type one prime are not equivalent, and investigates the properties of the type one prime radical. Several different types of conditions are given which guarantee that a prime ideal is type one. The class of all near-rings for which each prime ideal is type one is investigated and many examples of such near-rings are exhibited. Various localized distributivity conditions are found which are useful in establishing when prime ideals will be type one prime.     

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.     

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.     

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).     

A note on liftings of modules

Let N be a zero-symmetric right near-ring with identity. In 1993, S. Bagley introduced a construction for N[x], the near-ring of polynomials with coefficients from N. In this paper we study the central elements of N[x], C(N[x]), and we characterize C(N[x]) in terms of C(N) for a class of near-rings. We also introduce a new generalization for the center of a ring to the near-ring case, and we show that this new generalization yields a near-ring which properly contains C(N[x]) for a certain class of near-rings N.     

Dense Near-Rings of Operators on Locally Convex Spaces

We consider the near-ring C(V) of all continuous operators on a locally convex space V. Like in the Theorem of Stone-Weierstrass the question arises which subnear-rings N have the property that every operator in C(V) can be approximated by elements of N on compact subsets of V. It is our aim to show that this can be achieved with certain primitive subnear-rings of C(V). For this we invoke a deep Theorem of Wielandt-Betsch on interpolation properties of primitive near-rings. We also stress the fact that such a Theorem of Stone-Weierstrass type can only be obtained in the context of near-rings.     

Existence of derivations on near-rings

We obtain conditions on (R,+) which force that the zero map is the only derivation on a zero-symmetric near-ring R. Throughout the paper we construct several new examples of near-rings which are not rings admitting non-zero derivations, non-zero (σ, σ)-derivations and non-zero (1, σ)-derivations.     

Small resolutions and non-liftable Calabi-Yau threefolds

We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over \mathbbF₃{\mathbb{F}_3} that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over \mathbbF₅{\mathbb{F}_5} having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over \mathbbF_p{\mathbb{F}_p} that do not lift to algebraic spaces in characteristic zero.     

Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.