Nil Radical Research Articles

Modular Representation Theory and Commutative Banach Algebras

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to C \mathbb {C} . One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Pythagorean fuzzy nil radical of Pythagorean fuzzy ideal

In this work, we introduce the Pythagorean fuzzy nil radical of a Pythagorean fuzzy ideal of a commutative ring, we further provide the notion of Pythagorean fuzzy semiprime ideal, and we study some related properties. Finally, we give the relation between Pythagorean fuzzy semiprime ideals and the Pythagorean fuzzy nil radical of a commutative ring.

Some properties of neutrosophic nil radicals of neutrosophic ideals in rings

In this paper, we consider the notion of neutrosophic nil radicals of neutrosophic ideals in commutative rings and some properties of such nil radicals. Finally, we study the properties of semiprime neutrosophic ideals of rings.

Neutrosophic nil radicals of neutrosophic ideals in rings

The notion of neutrosophic nil radicals of neutrosophic ideals in rings is introduced, and related properties are investigated. In addition, we study some relations between the nil radical of the neutrosophic polynomial ideal of the polynomial ring R[x] induced by a neutrosophic ideal of a ring R and the nil radical of a neutrosophic ideal of R. Finally, we find the result of neutrosophic nil radicals of neutrosophic ideals from the epimorphism of rings.

Hybrid Nil Radical of a Ring

The nature of universe problems is ambiguous due to the presence of asymmetric data in almost all disciplines, including engineering, mathematics, medical sciences, physics, computer science, operations research, artificial intelligence, and management sciences, and they involve various types of uncertainties when dealing with them on various occasions. To deal with the challenges of uncertainty and asymmetric information, different theories have been developed, including probability, fuzzy sets, rough sets, soft ideals, etc. The strategies of hybrid ideals, hybrid nil radicals, hybrid semiprime ideals, and hybrid products of rings are introduced in this paper and hybrid structures are used to examine the structural properties of rings.

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

On intuitionistic nil radicals of an intuitionistic fuzzy ideals of hemirings

After the introduction of intuitionistic fuzzy sets by Atanassov K.T, many mathematical researchers have been devoted in the study of different aspects of this concept and they have been successful. The purpose of this paper is to introduce the notion of intuitionistic nil radicals of intuitionistic fuzzy ideals of hemirings. The author also investigated some of their properties.

SPECIAL RADICALS FOR NEAR-RING


 
 
 The concept of a special radical for near-rings has been treated in several nonequivalent, but related, ways in the recent literature. We use the version due to K. Kaarli to establish that various prime radicals and the nil radical are special radicals on the class A of all near-rings which satisfy an extended version of the Andrunakievich Lemma. Since A includes all d.g. near-rings—and much more—these results significantly extend results previously obtained by Kaarli and by Groenewald. We also obtain special radical results for the Jacobson type radicals 30 and 3 1 , albeit on less extensive classes. Examples are given which illustrate and delimit the theory developed. 
 
 

On Φ-projective modules and Φ-Prüfer rings

In this article, if S denotes a multiplicatively closed set of a commutative ring R, then S-fractional ideals and S-invertible ideals are introduced. Suppose that the nil radical Nil(R) of R is a divided prime ideal. Then ϕ-torsion free modules and ϕ-projective modules are introduced and it is shown that a finitely generated nonnil ideal I is ϕ-invertible if and only if I is ϕ-projective as R-module. So R is a ϕ-Prüfer ring if and only if each finitely generated nonnil ideal of R is ϕ-projective.

Radicals and Köthe’s Conjecture for Skew PBW Extensions

The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincare–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Kothe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

Five solved problems on radicals of Ore extensions

We answer several open questions and establish new results concerningdierential and skew polynomial ring extensions, with emphasis on radicals. In particular, we prove the following results. If R is prime radical and δ is a derivation of R, then the dierential polynomial ring R[X; δ] is locally nilpotent. This answers an open question posed in [41]. The nil radical of a dierential polynomial ring R[X; δ] takes the form I[X; δ] for some ideal I of R, provided that the base field is infinite. This answers an open question posed in [30] for algebras over infinite fields. If R is a graded algebra generated in degree 1 over a field of characteristic zero and δ is a grading preserving derivation on R, then the Jacobson radical of R is δ-stable. Examples are given to show the necessity of all conditions, thereby proving this result is sharp. Skew polynomial rings with natural grading are locally nilpotent if and only if they are graded locally nilpotent. The power series ring R[[X; σ; δ]] is well-defined whenever δ is a locally nilpotent σ-derivation; this answers a conjecture from [13], and opens up the possibility of generalizing many research directions studied thus far only when further restrictions are put on δ.

Normality on Jacobson and nil radicals

Normality on Jacobson and nil radicals

点态化完备代数正规类中的遗传幂等根、补根、对偶根、子幂等根及诣零根

点态化完备代数正规类中的遗传幂等根、补根、对偶根、子幂等根及诣零根

Another characterization of the upper nil radical

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 \({\rho}\) 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 \({\sigma}\)-ring. In this paper we introduce classes of rings equivalent to the \({\sigma}\)-rings and then give characterizations of the upper nil radical in terms of these rings.

Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ? N?.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 1-11

Radicals of Ore Extension of Skew Armendariz Rings

Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Niℓ*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results.

Veldsman’s classes of associative rings

The relation of being an ideal (left ideal, right ideal) of a ring is not transitive. Rings for which the transitivity does hold are called filial (left filial, right filial). In symbols, these are rings R such that if AαB and BαR then AαR, where α is the respective relation. In [24] Veldsman introduced for each triple (α, β, γ) of these relations the class of rings R which satisfy: if AαB and BβR then AγR. He found some relationships among so defined classes and characterized rings in each of them. Our aim in this paper is to study the structure of rings in these classes. We observe that one can select three from among them, which are fundamental in the sense that results obtained for them together with those known for left filial rings, give structure theorems for rings in all classes. Then we study the structure of rings in so selected three classes. Our main results concern the prime and generalized nil radicals of such rings and reduced rings in these classes. We also get many related results and ask several questions.

On the nil radical

Let \({\sigma}\) be the class of all associative rings 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 n, m such that i n j m = 0, then I = 0 or J = 0. We will show that \({\sigma}\) is a special class and the upper radical \({\mathcal{U}(\sigma)}\) determined by \({\sigma}\) is equal to the nil radical \({\mathcal{N}}\) of Koethe. This implies that the nil radical \({\mathcal{N}(R)}\) of any ring R is the intersection of all \({\sigma}\)-ideals I of R, that is, ideals I of R such that the factor ring R/I is in \({\sigma}\). We also give a necessary and sufficient condition for an ideal \({I \neq R}\) of a ring R to be a \({\sigma}\)-ideal.

Differential polynomial rings over rings satisfying a polynomial identity

Let R be a ring satisfying a polynomial identity and let δ be a derivation of R. We show that if R is locally nilpotent then R[x;δ] is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski. As a consequence we have that if R is a unital PI algebra over a field of characteristic zero then the Jacobson radical of R[x;δ] is equal to N[x;δ], where N is the nil radical of R.

ON THE LEVITZKI RADICAL OF MODULES

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 RM with RM 6= 0 forms a special class of modules.