Papers by nico groenewald
Bulletin of The Australian Mathematical Society, Dec 1, 2004
Polynomial near-rings in fc-commuting indeterminates are our object of study. We illustrate our w... more Polynomial near-rings in fc-commuting indeterminates are our object of study. We illustrate our work for k = 2, that is, iV[x,?/] as an extension to N[x], while the case for arbitrarily A; follows easily. Our approach is different from the recursive definition JV[x][i/]. However, it can be shown that N[x,y] is isomorphic to AT[x][y]. Several important tools such as the degree, the least degree, et cetera are defined with respect to N[x,y]. We also clarify some notations involved in defining polynomial neax-rings.
Publicationes Mathematicae Debrecen
Special radicals were defined for rings with involution by Salavov a. In this paper we show that ... more Special radicals were defined for rings with involution by Salavov a. 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 reverse inclusion does not hold in general. This theory gives new characterisations for certain concrete radicals.
Another characterization of the upper nil radical
Acta Mathematica Hungarica, Aug 23, 2016
In 2015 Halina France-Jackson introduced the notion of a $${\sigma}$$σ-ring i.e. a ring R with th... more In 2015 Halina France-Jackson introduced the notion of a $${\sigma}$$σ-ring i.e. a ring R with the property that if I and J are ideals of R and for all $${i\in I}$$i∈I, $${{j\in J}}$$j∈J, there exist natural numbers m, n such that $${i^{m}j^{n} =0}$$imjn=0, then I = 0 or J = 0. It is shown that $${\sigma}$$σ is a special class which coincides with the class $${\rho}$$ρ of all prime nil-semisimple rings. This implies that the upper nil radical of any ring R is the intersection of all ideals I of the ring such that R/I is a $${\sigma}$$σ-ring. In this paper we introduce classes of rings equivalent to the $${\sigma}$$σ-rings and then give characterizations of the upper nil radical in terms of these rings.
International Electronic Journal of Algebra, 2019
Let R be a noncommutative ring with identity. We define the notion of a 2-absorbing submodule and... more 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 proved which are similar to the results for commutative rings.
Hacettepe Journal of Mathematics and Statistics, 2016
We define and characterize classical completely prime submodules which are a generalization of bo... more We define and characterize classical completely prime submodules which are a generalization of both completely prime ideals in rings and reduced modules (as defined by Lee and Zhou in [18]). A comparison of these submodules with other "prime" submodules in literature is done. If Rad(M) is the Jacobson radical of M and β c cl (M) the classical completely prime radical of M , we show that for modules over left Artinian rings R, Rad(M) ⊆ β c cl (M) and Rad(RR) = β c cl (RR).
In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we de... more 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 define the Levitzki radical of an R-module M as L(M ) = l. √ 0. Let β(M ), U (M ) and Rad(M ) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M ) ⊆ L(M ) ⊆ U (M ) ⊆ Rad(M ). If R is commutative, β(M ) = L(M ) = U (M ) and if R is left Artinian, β(M ) = L(M ) = U (M ) = Rad(M ). Lastly, we show that the class of all l-prime modules R M with RM = 0 forms a special class of modules. Mathematics Subject Classification (2010):16D60, 16N40, 16N60, 16N80, 16N90
On the prime radicals of near-rings and near-ring modules
On the upper nil radical for near-ring modules
For a near-ring $R$ we introduce the notion of an $s-$prime $R-$module and an $s-$system. We show... more For a near-ring $R$ we introduce the notion of an $s-$prime $R-$module and an $s-$system. We show that an $R-$ideal $P$ is an $s-$prime $R-$ideal if and only if $R\backslash P$ is an $s-$system. For an $R-$ideal $N$ of the near-ring module we define $\mathcal{S}(N)=:\{s\in M~:~$every $s-$sytem containing $s$ meets $N\}$ and prove that it coincides with the intersection of all the $s-$prime $R-$ideals of $M$ containing $N.$ $S(0)$ is the upper nil radical of the near-ring module. Furthermore, we define a $\mathcal{T-}$% special class of near-ring modules and then show that the class of $s-$prime modules forms a $\mathcal{T-}$special class. $\mathcal{T-}$special classes of $s-$prime near-ring modules are then used to describe the upper nil radical of a near-ring
We define and characterize classical completely prime submodules which are a generalization of bo... more We define and characterize classical completely prime submodules which are a generalization of both completely prime ideals in rings and reduced modules (as defined by Lee and Zhou in [18]). A comparison of these submodules with other "prime" submodules in literature is done. If Rad(M) is the Jacobson radical of M and β c cl (M) the classical completely prime radical of M , we show that for modules over left Artinian rings R, Rad(M) ⊆ β c cl (M) and Rad(RR) = β c cl (RR).
On weakly 2-absorbing ideals of non-commutative rings
Afrika Matematika
Special radicals in rings with involution
Publicationes mathematicae
ABSTRACT
The completely prime radical in near-rings
Acta Mathematica Hungarica, 1988
Weakly Prime and Weakly Completely Prime Ideals of Noncommutative Rings
International Electronic Journal of Algebra
International Electronic Journal of Algebra
In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we de... more 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 define the Levitzki radical of an R-module M as L(M) = l. √ 0. Let β(M), U (M) and Rad(M) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M) ⊆ L(M) ⊆ U (M) ⊆ Rad(M). If R is commutative, β(M) = L(M) = U (M) and if R is left Artinian, β(M) = L(M) = U (M) = Rad(M). Lastly, we show that the class of all l-prime modules R M with RM = 0 forms a special class of modules.
Properties of Different Prime Radicals of Rings and Modules
Http Dx Doi Org 10 1080 00927872 2013 857237, Jan 7, 2015
ABSTRACT We investigate properties of different monoid module radicals arising from the different... more ABSTRACT 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 property; and if γ(M) = M, then γ(M(G)) = M(G), i.e., γ is polynomially extensible if M(G) = M[x]. We also show that a module R M is 2-primal if and only if the monoid module R(G)M(G) is 2-primal.
A generalization of prime ideals in near-rings
Communications in Algebra
A note on units and divisors of zero in group rings
Publications de l Institut Mathematique
Strongly semi-prime ideals in near-rings
Special radicals of Ω-groups
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Papers by nico groenewald