# Nico GroenewaldNelson Mandela University | NMMU · Department of Mathematics and Applied Mathematics

Nico Groenewald

Doctor of Philosophy

## About

95

Publications

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Introduction

Weakly prime ideals and weakly primary ideals in non commutative rings

## Publications

Publications (95)

In this study, we aim to introduce the concepts of 1-absorbing prime submodules and weakly 1-absorbing prime submodules of a unital module over a noncommutative ring with nonzero identity. This is a new class of submodules between prime submodules (weakly prime submodules) and 2-absorbing submodules (weakly 2-absorbing submodules). Let R be a nonco...

This paper introduce and study 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings a new class of ideals between prime ideals (weakly prime ideals) and 2-absorbing ideals (weakly 2-absorbing ideals). Let R be a noncommutative ring with a nonzero identity 1≠0. A proper ideal P of R is said to be a 1-absorbing prime i...

For a commutative ring [Formula: see text] denote the Jacobson radical of the ring by [Formula: see text]. Khashan et al. introduced and studied J-ideals and weakly J-ideals for commutative rings with identity. In this paper, let [Formula: see text] be a non-commutative ring. We introduce weakly [Formula: see text]-ideals for a special radical [For...

Let [Formula: see text] denote the Jacobson radical of a commutative ring [Formula: see text]. In [H. A. Khashan and A. B. Bani-Ata, [Formula: see text]-ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If [Formula: see text] denotes the prime radical of a commutative ring then in [U....

Let R be a commutative ring with identity element. The concept of 2-absorbing ideal was introduced by Badawi (Bull Aust Math Soc 75:417–429, 2007) as a generalization of the notion of prime ideal. Weakly prime ideals introduced by Anderson (Houst J Math 29(4):831–840 2003) are also generalizations of prime ideals. The concept of a weakly prime idea...

We introduce the notion of completely 2-absorbing (denoted by, c-2-absorbing) ideal of an N-group G, as a generalization of completely prime ideal of module over a right near-ring N. We obtain that, for an ideal I of a monogenic N-group G, if (I: G) is a c-2-absorbing ideal of N, then I is a c-2-absorbing ideal of G. The converse also holds only wh...

Anderson-Smith studied weakly prime ideals for a commutative ring with identity. Hirano, Poon and Tsutsui studied the structure of a ring in which every ideal is weakly prime for rings, not necessarily commutative. In this note we give some more properties of weakly prime ideals in noncommutative rings. We introduce the notion of a weakly prime rad...

Let R be a noncommutative ring with identity. We define the notion of a 2-absorbing submodule and show that if the ring is commutative then the notion is the same as the original definition of that of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are different. Many properties of 2-absorbing submodules are...

In 2015 Halina France-Jackson introduced the notion of a \({\sigma}\)-ring i.e. a ring R with the property that if I and J are ideals of R and for all \({i\in I}\), \({{j\in J}}\), there exist natural numbers m, n such that \({i^{m}j^{n} =0}\), then I = 0 or J = 0. It is shown that \({\sigma}\) is a special class which coincides with the class \({\...

Le Riche, Meldrum and Van der Walt introduced the concept of a group near-ring. Thereafter, starting with an ideal in the base near-ring, they constructed two corresponding ideals in the group near-ring. Furthermore, by starting with an ideal in the group near-ring, they constructed a corresponding ideal in the base near-ring. In this we paper, we...

We investigate properties of different monoid module radicals arising from the different definitions of “prime” modules. Let R be a unital ring, M an R-module, and G a monoid. If γ is a prime (resp. strongly prime and completely prime) radical of a monoid module M(G), then γ(M(G)) = γ(M)(G); (γ(M(G)) ∩ M)(G) = γ(M(G)), i.e., γ satisfies the Amitsur...

Öz
In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we define and characterize l-prime submodules. Let N be a submodule of an R-module M. If l.√N := {m ∈ M : every l- system of M containingm meets N}, we show that l.√N coincides with the intersection L(N) of all l-prime submodules of M containing N. We def...

We define nilpotent and strongly nilpotent elements of a module M and show that the set s (M) of all strongly nilpotent elements of M over a commutative unital ring R coincides with the classical prime radical βcl (M) the intersection of all classical prime submodules of M.

A notion of 2-primal rings is generalized to modules by defining 2-primal modules. We show that the implications between rings which are reduced, have insertion-of-factor-property (IFP), reversible, semi-symmetric and 2-primal are preserved when the notions are extended to modules. Like for rings, 2-primal modules bridge the gap between modules ove...

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define
$\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$
. It is shown that
$\mathcal{S}(N)$
is equal to the intersection of all s-prime submodules of M cont...

We generalize completely prime ideals in rings to submodules in modules. The notion of multiplicative systems of rings is generalized to modules. Let N be a submodule of a left R-module M. Define co. √ N := {m ∈ M : every multiplicative system containing m meets N }. It is shown that co. √ N is equal to the intersection of all completely prime subm...

Using Meldrum and van der Walt's scheme we successfully define (generalized) semigroup near-rings which are the extensions of their ring counterpart. Some standard semigroup ring results are extended. We define contracted objects for (generalized) semigroup near-rings and show (generalized) matrix near-rings are just a special case as in rings. Thi...

For near-ring ideal mappings ρ1 and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1: ρ2), and (ℛρ2: ℛρ1). Conditions on ρ1 and ρ2 are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to...

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between...

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.

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 investigat...

We introduce the notion of a strongly prime near-ring module and then characterize strongly prime near-rings in terms of strongly
prime modules. Furthermore, we define a T{\mathcal{T}}-special class of near-ring modules and then show that the class of strongly prime modules forms a T{\mathcal{T}}-special class. T{\mathcal{T}}-special classes of str...

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 investig...

It is well known that there are a number of different nonequivalent definitions of a prime near-ring. In this paper we show that this give rise to a number of different upper nil radicals which all coincide with the usual upper nil radical in the case of associative rings.

In this note we generalize the notion of 2-primal and nil-primal nearrings. For a Hoehnke radical ρ we introduce the notion of ρ–primal near-rings. 2-primal and nil-primal near-rings are examples of ρ–primal near-rings. We show that many of the results for 2-primal and nil-primal rings and near-rings can be proved for ρ–primal 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. © 2008 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou Univ...

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between...

In this paper we investigate the radical properties of classes of rings constructed using a class pair (M1 : M2) of rings. Our theory encompasses the theory of radical pairs and allows the study of ring theoretical questions in terms of radicals. For instance, Hilbert's Nullstellensatz and an equivalent form of Koethe's conjecture can be stated in...

Snider showed that the class of radicals of rings has a natural lattice structure. The same is true for any universal class of near-rings. We show that the classes of hereditary and ideal-hereditary radicals, inter alia, are complete sublattices. Atoms of certain sublattices are discussed. If N is a universal class of near-rings containing the univ...

We study prime rings which generate supernilpotent (respectively special) atoms, that is, atoms of the lattice of all supernilpotent (respectively special) radicals. A prime ring A is called a **-ring if the smallest special class containing A is closed under semiprime homomorphic images of A. A semiprime ring A whose every proper homomorphic image...

In this paper, we prove some basic properties of left weakly regular near-rings. We give an affirmative answer to the question whether a left weakly regular near-ring with left unity and satisfying the IFP is also right weakly regular. In the last section, we use among others left 0-prime and left completely prime ideals to characterize strongly re...

For near-ring ideal mappings ρ1 and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1 : ρ2), (Sρ2 : Sρ1) and (Rρ2 : R-ρ1). Conditions on ρ1 and ρ2 are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of exampl...

Polynomial near-rings in k-commuting indeterminates are our object of study. We illustrate out work for k = 2, that is, N[x, y] as an extension to N[x], while the case for arbitrarily k follows easily. Our approach is different from the recursive definition N[x][y]. However, it can be shown that N[x, y] is isomorphic to N[x][y]. Several important t...

A class K of rings has the GADS property (i.e., generalized ADS property) if wheneverX◃ I◃ R with X∈ K, then there exists B ◃ R with B ∈ K such that X ⊆ B ⊆ I. Radicals whose semisimple classes have the GADS property are called g-radicals. In this paper, we fully characterize the class of g -radicals. We show that ? is a g-radical if...

Let ℱ be a regularity for near-rings and ℱ(K) the largest F R-regular ideal in R. In the first part of this paper, we introduce the concepts of maximal F-modular ideals and F-primitive near-rings to characterize F̄(R) for any near-ring regularity ℱ. Under certain conditions, F̄(R) is equal to the intersection of all the maximal F-moduIar ideals of...

In this paper we introduce the concept of an almost nilpotent near-ring. We also define the almost nilpotent radical. The concept of a special radical for near-rings has been treated in several non-equivalent, but related ways in the recent literature. We use the version due to K. Kaarli as basis to define the concept of a weakly special radical in...

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We...

A lattice-theoretic approach to the radical theory of rings was initiated by Snider. In the current paper, we extend this approach to the radical theory of involution rings. We show that the classes of hereditary radicals, radicals satisfying ADS and invariant radicals form complete sublattices of the lattice of all radicals of involution rings. We...

An ideal A of a ring R is called a good ideal if the coset product r1r2 + A of any two cosets r1 + A and r2 + A of A in the factor ring R/A equals their set product (r1 + A) º (r2 + A): = {(r1 + a)(r2 + a2): a1, a2 ε A}. Good ideals were introduced in [3] to give a characterization of regular right duo rings. We characterize the good ideals of bloc...

In this paper, we introduce the concepts of right strongly semiprime and right semi-superprime near-rings. We show that a near-ring R is right strongly semiprime if and only if it is a subdirect product of finitely many strongly prime near-rings. We also show that, if R is a sero-symmetric near-ring, then the right strongly prime radical s(R) of R...

In this paper we investigate connections between the condition that every prime ideal is maximal and various generalizations of von Neumann regularity. As a corollary of our results we show that if N is a reduced zerosymmetric right near-ring, then every prime ideal is maximal if and only if N is left weakly regular (i.e., x∈〈x〉x, for all x∈N, wher...

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is...

. We investigate the relationships between the ideal structure and the -ideal structure of a ring with involution (). Descriptions of -minimal and - maximal ideals are obtained in terms of minimal and maximal ideals, respectively. Furthermore conditions are provided allowing us to associate with each minimal or maximal ideal a -minimal or -maximal...

In this paper we further demonstrate the breath of the abstract regularity concept for near-rings by introducing regularity-induced semiprime and prime conditions on ideals. We show that a link between an F-regular near-ring and an F-prime near-ring exists for a general class of near-ring regularities. Two main examples, f 0 -regularity and C-regul...

Buys and Gerber studied the theory of special radicals for An-drunakievich varieties of Ω-groups. We continue this study for not necessarily Andrunakievich varieties. Characterizations of the radical and semisimple classes are obtained, similar to those obtained for rings by Gardner and Wiegandt, and Rjabuhin and Wiegandt, respectively. These give...

In 1981 Van der Walt [12] asked: What is the strongest reasonable definition of a prime ring for non-commutative rings? The direct generalization from the commutative to the non-commutative case, namely completely prime, is too strong since for example, maximal ideals in a ring with identity need not be completely prime; no matrix ring can be compl...

Special radicals were defined for rings with involution by K. Salavovà [Commentat. Math. Univ. Carol. 18, 367-381 (1977; Zbl 0355.16003)]. In this paper we show that every special radical ℛ in the variety of rings induces a corresponding special radical ℛ * in the variety of rings with involution, and ℛ * (R)⊆ℛ(R) for any involution ring R. The rev...

In this paper we consider a class of Hoehnke radicals of associative rings. A radical H from this class is constructed using generalized semi-prime ideals determined by a ring regularity. It is shown that different regularitiesMaybe used to construct the same radical H. Furthermore, we show that the K.A.-radical corresponding to one of the regulari...

Let M be a F-ring with left and right operator rings L and R respectively. It was previously shown that if ~ is an N-radical class of rings, then ~(L) + = 7~(R)*. In the current paper, we extend this result to the wider class of normal radicals. It is also shown that, if ~ is a supernilpotent radical class of rings and "~(L) + = ~(R)* for every F-r...

We establish a link between the two characterizations, independently obtained in 1991 by Puchs and Robson, of complete matrix rings in terms of the existence of nilpotent elements.

It is known that many special radicals R satisfy the matrix condition an arbitrary ring, but that these radicals do not carry over from the base ring R to structural matrix rings in this natural way. In this paper we study the radicals (of structural matrix rings) determined by some polynomial regularities. We obtain various classes of polynomial r...

A general regularity for weak Nobusawa γ-rings is defined which takes into account the nature of the factor r-rings in this category. The class of *-regularities is defined and studied. We show that *-regularities represent only six different algebraic properties for elements of γ-rings. Right (left weak-regularity, Von Neumann-regularity, α-regula...

Let R be a zerosymmetric near-ring with identity. The near-ring M m (R) of m×m matrices over R was recently defined by Meldrum and Van der Walt. We define a Γ-near-ring of m×n matrices over R, with left operator near-ring L. Various relationships are established between certain classes of ideals of R, L, and M. The equiprime, strongly equiprime and...

Using a general definition of a regularity for rings, F- and F- qausi-ideals of a ring are defined. These concepts are shown to be generalizations of ideals or one-sided ideals of a ring. An F-semi prime F—(F-quasi-) ideal of a ring R is also defined. F-regular rings are characterized in terms of F-semi prime F- (F-quasi-) ideals for a large class...

Let M be a Γ-ring with right operator ring R . We define one-sided ideals of M and show that there is a one-to-one correspondence between the prime left ideals of M and R and hence that the prime radical of M is the intersection of its prime left ideals. It is shown that if M has left and right unities, then M is left Noetherian if and only if ever...

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...

We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only...

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 compara...

The object of this paper is to define a general regularity for Γ-rings and to explore ways of generating such regularities. Two main approaches are identified. Regularities for Γ-rings can be generated by means of polynomials. These methods are the main topic of Section 2 where the concepts of polynomial, ideal generated polynomial and ideal genera...

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...

The purpose of this paper is to generalize the concept of semi prime ideals in GAMMA-rings. We use a general definition of a regularity F for GAMMA-rings to define an F-semi prime ideal. Relationships between F-semi prime ideals of a GAMMA-ring M and F-semi prime ideals of the operator rings R and L are discussed. D-regularity, f-regularity and lam...

In this paper we introduce the concept of almost nilpotence for Γ-rings, similar to the corresponding concept for rings, as defined by Van Leeuwen and Heyman. An almost mlpotent radical property Α0 is introduced for Γ-rings, and shown to be supernilpotent. If M is a Γ-ring with left and right operator rings L and R respectively, then Α(L)+ = Α0(M)...

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 a...

We introduce the notion of semi-uniformly strongly prime near-rings. A notion that coincides with the notion of a 3-semiprime near-ring in the case of finite near-rings. It is still an open question as to whether the 3-prime and 3-semiprime near-rings give rise to the same radical. In this note we show that the uniformly strongly prime and semi-uni...

Certain regularities in associative rings have been known for a long time. Most of these regularities have since been defined for near- rings, as is evident from the examples in paragraph 4. In this paper we shall attempt to develop a general theory of regularities for near- rings. Boos managed to define the concept of a regularity for rings in gen...

Strongly prime rings were introduced by Handelman and Lawrence [ 5 ] and in [ 2 ] Groenewald and Heyman investigated the upper radical determined by the class of all strongly 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 class in the sen...

In this note a completely prime radical is defined for near-rings. It is then proved that for the class of all distributively generated near-rings the completely prime radical coincides with the generalized nil radical. We also show that in general the completely prime radical and also the prime radical is not hereditary.

The concept of uniformly strongly prime (usp) is introduced for Γ-ring, and a usp radical τ(M) is defined for a Γ-ring M. If M has left and right unities, then τ(L)+ = τ(M) = τ(R)*, where L and R denote, respectively, the left and right operator rings of M, and τ(·) denotes the usp radical of a ring. If m, n are positive integers, then τ(Mmn) = (τ(...

In this note we introduce the concept of completely prime and completely semiprime ideals for Γ-rings and study some of the properties of these ideals. Furthermore we introduce the concept of the completely prime radical of an ideal of a Γ-ring and also show that the upper radical determined by a class of completely prime rings is a special radical...

It is known that in a near-ring N the Levitzki radical L(N), that is, the sum of all locally nilpotent ideals, is the intersection of all the prime ideals P in N such that N/P has zero Levitzki radical. The purpose of this note is to prove that L(N) is the intersection of a certain class of prime ideals, called l-prime ideals. Every l-prime ideal P...

(Opgedra aan Prof. Hennie Schutte by geleentheid van sy sestigste verjaarsdag.)AbstractIn this note we introduce the concept of a special class of Γ-rings. The upper radical property determined by a special class is then called a special radical. We show that the prime radical, Levitzki nil radical and the nil radical are all special radicals. Next...

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is...

In this paper we show that the definition and construction of radical and semisimple classes of associative rings can be interpreted in a general category K in terms of two subclasses of epimorphisms and mono-morphisms. We also provide answers to the following two questions posed by Wiegandt:1. Which objects should be excluded when defining radical...

Let R be a ring with identity and H a normal subgroup of the group G. In this paper the relationship between certain classes of ideals in RH (or R) and the group ring RG is investigated by employing McCoy's “going up” and “going down” method which he used for polynomial rings in [2]. From the results obtained it is inferred that P(R)S = P(RS) if S...

In [2] Coleman and Enochs obtained results about the units of the polynomial ring R[x] for rings R satisfying a condi-tion which is, in some sense, a generalization of commutativity. In [3] some of these results were extended to group rings over an ordered group. In this note a class of rings larger than the class considered in [2] is used to exten...

If R is a ring such that x, y ∈ R and xy = 0 imply yx = 0 and G≠ 1, an ordered group, then we show that ∑ α g g is a unit in RG if and only if there exists ∑ β h h in RG such that ∑ α g β g-1 = 1 and α g β h is nilpotent whenever gh≠l. We also show that if R is a ring with no nilpotent elements ≠ 0 and no idempotents ≠ 0, 1 then RG has only trivial...

Let R be a ring with identity and H a normal subgroup of the group G. In this paper the relationship between certain classes of ideals in RH (or R) and the group ring RG is investigated by employing McCoy's “going up” and “going down” method which he used for polynomial rings in [2]. From the results obtained it is inferred that P(R)S = P(RS) if S...

A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note...