Zero-symmetric Near-ring Research Articles

Superfluous ideals of N-groups

We consider a right nearring N and a module over N (known as, N-group). For an arbitrary ideal (or N-subgroup) varOmega of an N-group G, we define the notions varOmega -superfluous, strictly varOmega -superfluous, g-superfluous ideals of G. We give suitable examples to distinguish between these classes and the existing classes studied in Bhavanari (Proc Japan Acad 61-A:23–25, 1985; Indian J Pure Appl Math 22:633–636, 1991; J Austral Math Soc 57:170–178, 1994), and prove some properties. For a zero-symmetric nearring with 1, we consider a module over a matrix nearring and obtain one-one correspondence between the superfluous ideals of an N-group (over itself) and those of M_{n}(N)-group N^{n}, where M_{n}(N) is the matrix nearring over N. Furthermore, we define a graph of superfluous ideals of a nearring and prove some properties with necessary examples.

Nearrings of functions without identity determined by a single subgroup

Let $(G, +)$ be a finite group, written additively with identity 0, but not necessarily abelian, and let $H$ be a nonzero, proper subgroup of $G$. Then the set $M = {f : G to G | f(G) subseteq H hbox{and} f(0) = 0 }$ is a right, zero-symmetric nearring under pointwise addition and function composition. We find necessary and sufficient conditions for $M$ to be a ring and determine all ideals of $M$, the center of $M$, and the distributive elements of $M$.

On clean and regular elements of near-ring of skew polynomials

In this paper, we are interested to investigate some properties of the zero-symmetric near-ring of skew polynomials We first show that if R is an -compatible ring and Nil(R) is a locally nilpotent ideal of R, then the set of all nilpotent elements of the near-ring forms an ideal and Then we characterize the unit elements, the regular elements, the π-regular elements and the clean elements of where the base ring R is a semicommutative ring with Also, we prove that the set of all π-regular elements of forms a semigroup. These results are somewhat surprising since, in contrast to the skew polynomial ring case, the near-ring of skew polynomials has substitution for its “multiplication” operation.

On quasi-radical of near-ring of polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

A Characterization of Left Regularity

We show that a zero-symmetric near-ring $N$ is left regular if and only if $N $ is regular and isomorphic to a subdirect product of integral near-rings, where each component is either an Anshel-Clay near-ring or a trivial integral near-ring. We also show that a zero-symmetric near-ring is regular without nonzero nilpotent elements if and only if the multiplicative semigroup of N is a union of disjoint groups.

Finite local nearrings

Nearrings arise naturally in the study of systems of nonlinear mappings, and they have been studied for many decades. Basic definitions and many results concerning nearrings can be, for instance, found in [G. Pilz. Near-rings. The theory and its applications. North Holland, Amsterdam, 1977].Nearrings are generalized rings in the sense that the addition need not be commutative and only one distributive law is assumed. Clearly, every associative ring is a nearring, and each group is an additive group of a nearring, but not necessarily of a nearring with identity. The question what group can be an additive group of a nearring with identity is far from solution.A nearring with identity is called local if the set of all its non-invertible elements is a subgroup of its additive group. A study of local nearrings was initiated by Maxson (1968) who defined a number of their basic properties and proved, in particular, that the additive group of a finite zero-symmetric local nearring is a p-group. The determination of the non-abelian finite p-groups which are the additive groups of local nearrings is an open problem (Feigelstock, 2006).The list of all local nearrings of order at most 31 can be extracted from the package SONATA (https://www.gap-system.org/Packages/sonata.html) of the computer system algebra GAP (https:// www.gap-system.org/).We observe also that there exist 14 non-isomorphic groups of order 16 = 24 from which 9 are the additive groups of local nearrings. Groups of order 32 = 25 with this property are described. In particular, among 51 non-isomorphic groups of this order only 19 are these additive groups.In this paper finite local nearrings are studied. Moreover, local nearrings of order at most 32 are classified.

On two sided α-n-derivation in 3-prime near-rings

Let N be a left near-ring and let α be a function of N. We introduce the notion of two sided α-n-derivation and prove that a prime zero symmetric near-ring involving α-n-derivations satisfying certain identities is a commutative ring.Also, some examples are given to shown that the 3-primeness condition in our results is not redundant.

Nearring ideals, graphs and cliques

In this paper, we dene the notion of fuzzy graph of a nearring N with respect to a level ideal t denoted by (N;;;t ). The primary aim of this notion is to depict graphically the fuzzy character which is concealed algebraically in the examples of 3-prime fuzzy ideals of a nearring N. We nd that if N is a zero-symmetric nearring and is 3prime fuzzy ideal of N, then (N;;;t ) has a special type of symmetry. We call this symmetry as the ideal symmetry of (N;;;t ). We nd the conditions under which the ideal symmetry of (N;;;t ) implies is a 3-prime fuzzy ideal of N. Finally, we obtain a result which nds all the fuzzy cliques of (N;;;t ) whenever is 3-prime and N is zerosymmetric.

Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring \({\mathbb{M}_n(R) (n >1 )}\) and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \({\mathbb{M}_n(R)}\) has the descending chain condition on \({\mathbb{M}_n(R)}\)-subgroups, then the 0-primitivity of \({\mathbb{M}_n(R)}\) implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \({\mathbb{M}_n(R)}\) is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.

Different Prime R-Ideals

It is well known that there are several non-equivalent types of prime near-rings which are all equivalent in the case of associative rings. In this paper we introduce various characterizations of prime modules in a zero-symmetric near-ring R. The connection of a prime R-ideal P of a module M and the ideal (P:M) of the near-ring R is also investigated.

Graph of a Nearring with Respect to an Ideal

We introduce a concept called the graph of a nearring N with respect to an ideal I of N denoted by G I (N). Then we define a new type of symmetry called ideal symmetry of G I (N). The ideal symmetry of G I (N) implies the symmetry determined by the automorphism group of G I (N). We prove that if I is a 3-prime ideal of a zero-symmetric nearring N then G I (N) is ideal symmetric. Under certain conditions, we find that if G I (N) is ideal symmetric then I is 3-prime. Finally, we deduce that if N is an equiprime nearring then the prime graph of N is ideal symmetric.

Compatible extensions of nearrings

If R is a zero-symmetric nearring with 1 and G is a faithful R-module, a compatible extension of R is a subnearring S of M 0(G) containing R such that G is a compatible S-module and the R-ideals and S-ideals of G coincide. The set of these compatible extensions forms a complete lattice and we shall study this lattice. We also will obtain results involving the least element of this lattice related to centralizers and the largest element of this lattice related to uniqueness of minimal factors with an application to 1-affine completeness of the R-module G.

Prime Ideals in Group Near-rings

The flow (or lack thereof) of several kinds of primeness between a zero-symmetric near-ring R and its group near-ring R[G] for certain groups G is discussed. In certain cases, results are contrasted against what happens in the matrix near-ring situation.

Local nearrings with generalized quaternion multiplicative group

A not necessarily zero-symmetric nearring R with identity is called local if the set of all non-invertible elements of R forms a subgroup of its additive group. The local nearrings whose multiplicative group is generalized quaternion are described. In particular, it is proved that their additive groups are abelian of types (3,3), (2,2,2,2), (2,2,4), (2,2,2,2,2) and (2,2,2,4). Keywords: Local nearring, Multiplicative group, Generalized quaternion group, Factorized group Mathematics Subject Classification (2000): 16N20, 16U60, 20M25

A BOUND ON THE NUMBER OF UNARY POLYNOMIAL FUNCTIONS AND A DECIDABILITY RESULT FOR NEAR-RINGS

Given a finite zero-symmetric near-ring with identity N, we ask whether there is a group G such that N is isomorphic to the inner automorphism near-ring <I(G);+,◦>, or whether N is a compatible near-ring. We will show that there are algorithms that decide these questions. To this end, we study polynomial functions on subdirectly irreducible expanded groups. We prove that the size of a finite subdirectly irreducible expanded group is bounded from above by a function of the number of its zero-preserving unary polynomial functions.

Local nearrings with dihedral multiplicative group

A not necessarily zero-symmetric nearring R with a unit element is called local if the set of all non-invertible elements of R forms a subgroup of the additive group of R. It is proved that every local nearring whose multiplicative group is dihedral is finite and its additive group is either a 3-group of order at most 9 or a 2-group of order at most 32.

ON NEAR-RING IDEMPOTENTS AND POLYNOMIALS ON DIRECT PRODUCTS OF Ω-GROUPS

AbstractLet $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

A NEW GENERALIZATION OF THE CENTER OF A NEAR-RING WITH APPLICATIONS TO POLYNOMIAL NEAR-RINGS

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.

Chains of intermediate ideals in matrix near-rings

0. Introduction. It is well-known that there is a one to one correspondence between the set of(two-sided) ideals of a ring R with identity and the set of ideals of the corresponding matrix ring ~ (R), for any n e N. Moreover, the correspondence appears in a very trivial and natural way, as the reader would have seen in many a textbook. Since the introduction of matrices over arbi trary near-rings by Meldrum and van der Walt ([4]) in /984, ideals in the near-ring R and related ideals in the corresponding matrix near-ring JM~ (R) have been extensively studied. In 1987 van der Walt ([7]) has shown that there are in general more ideals in a matrix near-ring than in its base near-ring. He did this by relating ideals in the matrix near-ring to a given ideal in the base near-ring in two natural ways. These two methods coincide when the base near-ring is a ring, but not in the general near-ring setting, as an example in [7] shows. This was the first evidence of the preponderance of ideals in 1M~, (R) over those in R. Since then, more examples of this kind were constructed, even for near-rings which are "very nearly" rings. (See [t] and [3]). In a recent investigation by Meldrum and the author (see [2]), this process is t aken a step further: we have shown that, apart from the additional ideal in ]M, (R) that one can expect by performing the two natural constructions discussed in the previous paragraph on an ideal in R, there are, in general, even more ideals the so-called intermediate ideals lurking around in a matrix near-ring. One of the examples given in [2] was a zero-symmetric abelian near-ring and it was conjectured that arbitrarily long chains of intermediate ideals can be constructed by using this example. In this note it is shown that such chains of intermediate ideals do indeed exist.

Strongly prime near-rings 2

by N. J. GROENEWALD(Received 24th October 1985, revised 22nd July 1987)1. IntroductionStrongly prime rings were introduced by Handelman and Lawrence [5] and in [2]Groenewald and Heyman investigated the upper radical determined by the class of allstrongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special classin the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincideswith the intersection of all the strongly prime ideals of N.2. PreliminariesUnless otherwise stated, all near-rings are zero-symmetric right near-rings. Forundefined terminologies, we refer to [9].Definition 1. Let N be a near-ring. N {right) is calle stronglyd prime if and only if forevery O^aeN there exists a finite subset F of N r(aF) such = tha {neN:aFn=O}t = 0. Fis called an insulator of a in N.We now give the following alternative definition (c.f. [8] for corresponding definitionfor rings).Definition 2. Let N be a near-ring. N is called (right) strongly prime if and only ifevery nonzero ideal I of N contains a finite subset F 0. such that r(F) =The two definitions of strongly prime agree for the class of zero-symmetric near-rings.The proof of this is based on the following observation and lemma.Observation. Let N be a zero-symmetric near-ring and XcN. The ideal generatedby X is the intersection of all ideals containing X and can be obtained as follows: