Properties Of Near-rings Research Articles

On some properties of near-rings

We establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d, we investigate conditions such as d([U,V])subseteq Z(R), d(U)subseteq Z(R), x_{o}d(R)subseteq Z(R), and Ux_{o}subseteq Z(R). As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if xin Z(R), then d(x)in Z(R). In addition, we provide useful examples illustrating our results.

SOME PROPERTIES OF (m, n)-POTENT CONDITIONS

In this paper, we will consider the notions of (m, n)-potent conditions in near-rings, in particular, a near-ring R with left bipotent or right bipotent condition. We will derive some properties of near-rings with (1, n) and (n, 1)-potent conditions where n is a positive integer, and then some properties of near-rings with (m, n)-potent conditions. Also, we may discuss the behavior of R-subgroups in (1, n)-potent or (n, 1)-potent near-rings..

SOME ISOMORPHIC PROPERTIES OF NEAR-RINGS

In this paper, we denote that is a near-ring and an -group. We initiate the study of faithful -group, the substructures of and , also quotients of substructure relations between them. Next, we investigate some isomorphic properties of near-rings and -groups.

Some properties of linear right ideal nearrings

In a previous paper, we determined all those topological nearrings 𝒩n whose additive groups are the n‐dimensional Euclidean groups, n > 1, and which contain n one‐dimensional linear subspaces which are also right ideals of the nearring with the property that for each w ∈ 𝒩n, there exist wi ∈ Ji, 1 ≤ i ≤ n, such that w = w1 + w2 + ⋯+wnand vw = vwn for each v ∈ 𝒩n. In this paper, we determine the properties of these nearrings, their ideals, and when two of these nearrings are isomorphic, and we investigate the multiplicative semigroups of these nearrings.

Intermediate ideals in matrix near-rings

One of the ways in which matrix near-rings as defined by Meldrum and van der Walt differ from matrix rings is that the correspondence between ideals in the matrix near-ring and those in the base near-ring is many-one, unlike the ring case. Here it is shown that, in suitable cases, an arbitrarily large lattice of ideals in the matrix near-ring can be made to correspond to an ideal in the base near-ring. This is achieved in two quite different contexts, using different properties of near-rings. The relationship between the ideals in the matrix near-ring and those in the base near-ring is also examined in some detail. Matrix near-rings as defined by Meldrum and van der Walt [7] have proved to be a fruitful concept. One of the ways in which they behave differently from matrix rings is that the two most obvious ways of extending an ideal from the base near-ring to the matrix near-ring do not always yield the same object, unlike the situation with rings. This leads to the idea of an intermediate ideal, one that lies between the two extensions of the ideal in the base near-ring. We show here that such ideals exist in abundance. This is done in two very different contexts. The first context is that of polynomial near-rings, whose addition is commutative, and where it is the multiplicative structure which is the key to the construction. The second context is that of d. g. weakly distributive near-rings, where the additive structure plays an important role and the multiplicative structure is very simple. We also point out some elementary properties of intermediate ideals before starting on the details of the constructions. But first an introduction to the whole subject.

Projective Klingenberg planes over nearrings

In the first section the definition of a PK-nearring is given and it is shown that every PK-nearring is a local ternary ring. Then a PK-nearring is used to construct PK-planes and further algebraic properties of PK-nearrings are related to geometrical ones of the corresponding PK-planes. Finally, a wide class of PK-nearrings is presented by applying the Dickson-process to the ring of formal power series. Throughout the following all nearrings shall be left-distributive. For the notation and basic properties of nearrings the reader is referred to Pilz [5].

On Near-Rings of Quotients

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank