Zero-symmetric Near-rings Research Articles

A Study on n-Derivation in Prime Near – Rings

The main purpose of this paper is to show that zero symmetric prime near-rings, satisfying certain identities on n-derivations, are commutative rings.

On additively completely regular seminearrings: II

As a continuation of our previous work, an attempt has been made to obtain some sort of analog of structure theorem of Clifford semigroups in the setting of seminearrings. To accomplish this, the notion of strong bi-semilattice of seminearrings has been introduced. Then those left (right) Clifford seminearrings, which are strong bi-semilattice of near-rings (zero-symmetric near-rings) and strong distributive lattice of near-rings (zero-symmetric near-rings) have been characterized.

A note on additively completely regular seminearrings

In the endeavour of obtaining semigroup theoretic analogues i.e., the analogues of structure theorems of completely regular semigroups in the setting of additively regular seminearrings we could obtain some results in Mukherjee et al. (Commun Algebra 45(12):5111–5122, 2017). But we could not obtain the analogue of (i) ‘A semigroup is Clifford if and only if it is strong semilattice of groups’ and (ii) ‘ A semigroup is completely regular if and only if it is a union of groups’. In Mukherjee et al. (Commun Algebra, https://doi.org/10.1080/00927872.2018.1524011) we could obtain the analogue of (i) for some restricted type of left (right) Clifford seminearrings. The main purpose of this paper is to complete the remaining task i.e., to obtain the analogue of (ii) for a class of additively completely regular seminearrings. In order to accomplish this we have characterized those seminearrings which are union of near-rings (zero-symmetric near-rings) in the class of additively completely regular seminearrings.

On additively completely regular seminearrings

ABSTRACTIn an attempt to investigate the situation arising out of replacing additive regularity by additive complete regularity in our previous study on additively regular seminearrings, we introduce the notions of left (right) completely regular seminearrings and characterize left (right) completely regular seminearrings as bi-semilattices of left (resp., right) completely simple seminearrings. We also define left (right) Clifford seminearrings and show that they are precisely bi-semilattices of near-rings (resp., zero-symmetric near-rings).

Hereditary right Jacobson radicals of type-1(e) and 2(e) for right near-rings

Abstract Near-rings considered are right near-rings. In this paper two more radicals, the right Jacobson radicals of type-1(e) and 2(e), are introduced for near-rings. It is shown that they are Kurosh-Amitsur radicals (KAradicals) in the class of all near-rings and are ideal-hereditary radicals in the class of all zero-symmetric near-rings. Different kinds of examples are also presented.

A KUROSH-AMITSUR LEFT JACOBSON RADICAL FOR RIGHT NEAR-RINGS

Let R be a right near-ring. An R-group of type-5/2 which is a natural generalization of an irreducible (ring) module is introduced in near-rings. An R-group of type-5/2 is an R-group of type-2 and an R-group of type-3 is an R-group of type-5/2. Using it <TEX>$J_{5/2}$</TEX>, the Jacobson radical of type-5/2, is introduced in near-rings and it is observed that <TEX>$J_2(R){\subseteq}J_{5/2}(R){\subseteq}J_3(R)$</TEX>. It is shown that <TEX>$J_{5/2}$</TEX> is an ideal-hereditary Kurosh-Amitsur radical (KA-radical) in the class of all zero-symmetric near-rings. But <TEX>$J_{5/2}$</TEX> is not a KA-radical in the class of all near-rings. By introducing an R-group of type-(5/2)(0) it is shown that <TEX>$J_{(5/2)(0)}$</TEX>, the corresponding Jacobson radical of type-(5/2)(0), is a KA-radical in the class of all near-rings which extends the radical <TEX>$J_{5/2}$</TEX> of zero-symmetric near-rings to the class of all near-rings.

Structure Theorems for the Socle-Ideal of a Near-Ring

For d.g. near-rings R satisfiying the descending chain condition for left ideals, Laxton and Machin (1972) used a faithful R-group to construct an ideal, which they called the critical ideal. The critical ideal turns out to be the socle-ideal of the d.g. near-ring R and its construction is based on the classification of the isomorphism classes of R-groups of type-0. We consider more general zero symmetric near-rings and although our method of classification is quite different from the Laxton–Machin one, we end up with the same isomorphism classes of R-groups of type-0. The socle-ideal is realized as a direct sum of copies of R-groups of type-0 from one of the classes. We then apply our theory to factor near-rings arising from nil-rigid series and to matrix near-rings.

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.

Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

The Variety of Near-Rings is Generated by Its Finite Members

We show that the variety of near-rings and the variety of zero-symmetric near-rings are both generated by their finite members. We show this in a more general context: if a variety \({\cal V}\) is generated by a class of algebras \({\cal F}\), then the variety of \({\cal V}\)-composition algebras is generated by the class of all full function algebras on direct products of finitely many copies of algebras in \({\cal F}\).

Nearrings whose set of N-subgroups is linearly ordered

We consider zerosymmetric right nearrings whose lattice of N-subgroups is linearly ordered by inclusion and in which for every n ∈ N, there is x ∈ N such that x * n = n. All such nearrings with finitely many N-subgroups are constructed.

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical\(\mathcal{G}\) is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals,\(\mathcal{B}\) and\(\mathcal{S}\), respectively, for zerosymmetric Γ-near-rings. Both\(\mathcal{B}\) and\(\mathcal{S}\) are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric Γ-near-ring with left operator near-ringL, it is shown that\(\mathcal{G}\left( L \right)^ + \subseteq \mathcal{B}\left( M \right)\), with equality ifM has a strong left unity.\(\mathcal{G}\) is extended to the variety of arbitrary near-rings, and\(\mathcal{B}\) and\(\mathcal{S}\) are extended to the variety of arbitrary Γ-near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA ⊲N, then\(\mathcal{G}\left( A \right) \subseteq A \cap \mathcal{G}\left( N \right)\), with equality ifA if left invariant.

A kurosh-amitsur prime radical for near-rings

The usual definition of primeness for near-rings does not lead to a Kurosh-Amitsur radical class for zerosymmetric near-rings (cf. Kaarli and Kriis[5]). In this paper, we define equiprime near-rings, which are another generalization of primeness in rings. Various results are proved, amongst others, for zerosymmetric near-rings:If N is a near-ring and A ⊲ B ⊲ N such that B/A is equiprime, then A ⊲ N. We define . Then p∗ is a Kurosh-Amitsur radicial for which holds for all ideals I of N. Moreover , where j 3 is the Jacobson-type radical which was introduced by Holcombe, with equality if N has the descending chain condition on N-sub-groups.

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:

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by Scukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.