Nil Radical Research Articles

A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

On Dixmier–Duflo isomorphism in positive characteristic – The classical nilpotent case

Let g be the nil radical of the Borel subalgebra of one of the classical simple Lie algebras over a field F of characteristic p⩾0. For p>0 we find an explicit realization of the center Z(g) of the enveloping algebra U(g) by generators and relations. This constructive approach yields an explicit isomorphism between Z(g) and the polynomial invariants algebra S(g)g. While realizing Z(g), we also prove that Z(g) is a complete intersection ring. Moreover, it leads to an explicit realization of Z(g) and S(g)g for p=0 as well. This extends a result of Dixmier in type An.

RADICALS OF SKEW GENERALIZED POWER SERIES RINGS

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a 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 generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.

Proper Fuzzification of Prime Ideals of a Hemiring

Prime fuzzy ideals, prime fuzzyk-ideals, and prime fuzzyh-ideals are roped in one condition. It is shown that this way better fuzzification is achieved. Other major results of the paper are: every fuzzy ideal (resp.,k-ideal,h-ideal) is contained in a prime fuzzy ideal (resp.,k-ideal,h-ideal). Prime radicals and nil radicals of a fuzzy ideal are defined; their relationship is established. The nil radical of a fuzzyk-ideal (resp., anh-ideal) is proved to be a fuzzyk-ideal (resp.,h-ideal). The correspondence theorems for different types of fuzzy ideals of hemirings are established. The concept of primary fuzzy ideal is introduced. Minimum imperative for proper fuzzification is suggested and it is shown that the fuzzifications introduced in this paper are proper fuzzifications.

Principally Small Injective Rings

Abstract. A right ideal I of a ring R is small in case for every proper right ideal Kof R, K + I 6= R. A right R-module M is called PS-injective if every R-homomorphismf : aR ! M for every principally small right ideal aR can be extended to R ! M. A ringR is called right PS-injective if R is PS-injective as a right R-module. We develop, inthis article, PS-injectivity as a generalization of P-injectivity and small injectivity. Manycharacterizations of right PS-injective rings are studied. In light of these facts, we getseveral new properties of a right GPF ring and a semiprimitive ring in terms of rightPS-injectivity. Related examples are given as well. 1. IntroductionThroughout this paper, R is an associative ring with identity and all modulesare unitary. Let R be a ring. The Jacobson radical and nil radical of R are denotedby J(R) and Nil(R), respectively. The right singular ideal is denoted by Z(R R ),the socles are denoted by soc(R R ) and soc( R R). If X is a subset of R, the right(resp. left) annihilator of X in R is denoted by r

On the Fuzzy Nil Radicals of Fuzzy Ideals of κ-semirings

On the Fuzzy Nil Radicals of Fuzzy Ideals of κ-semirings

Partial Actions and Partial Fixed Rings

In this article, we consider a partial action α of a finite group G on a ring R which possesses an enveloping action. Properties of the fixed subring R α are studied. Among other results, under some assumptions, we describe the prime radical, the Jacobson radical, and the upper nil radical. We also study semisimplicity, von Newmann regularity, and Goldie conditions.

THE IDEALS OF AN IDEAL EXTENSION

Given two unital associative rings R ⊆ S, the ring S is said to be an ideal (or Dorroh) extension of R if S = R ⊕ I, for some ideal I ⊆ S. In this note, we investigate the ideal structure of an arbitrary ideal extension of an arbitrary ring R. In particular, we describe the Jacobson and upper nil radicals of such a ring, in terms of the Jacobson and upper nil radicals of R, and we determine when such a ring is prime and when it is semiprime. We also classify all the prime and maximal ideals of an ideal extension S of R, under certain assumptions on the ideal I. These are generalizations of earlier results in the literature.

Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings

We first study connections between α‐compatible ideals of R and related ideals of the skew Laurent polynomials ring R[x, x−1; α], where α is an automorphism of R. Also we investigate the relationship of P(R) and Nr(R) of R with the prime radical and the upper nil radical of the skew Laurent polynomial rings. Then by using Jordan′s ring, we extend above results to the case where α is not surjective.

On the Commutative Twisted Group Algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ℬ(R) and ℬ(R t G) the nil radicals of R and R t G, respectively, by G p the p-component of G and by G 0 the torsion subgroup of G. We prove that: i. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ℬ(R) = 0, then there exists a twisted group algebra R t 1 (G/G p ) such that R t G/ℬ(R t G) ≅ R t 1 (G/G p ) as R-algebras; ii. If R is a ring of prime characterisitic p and R* is p-divisible, then ℬ(R t G) = 0 if and only if ℬ(R) = 0 and G p = 1; and iii. If B(R) = 0, the orders of the elements of G 0 are not zero divisors in R, H is any group and the commutative twisted group algebra R t G is isomorphic as R-algebra to some twisted group algebra R t 1 H, then R t G 0 ≅ R t 1 H 0 as R-algebras.

Different nil radicals for near-rings

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.

Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings

The notion of intuitionistic nil radicals of intuitionistic fuzzy ideals in rings is introduced, and related properties are investigated. The notion of semiprime intuitionistic fuzzy ideals is provided, and its characterization is established. The concept of Euclidean intuitionistic fuzzy ideals is also introduced, and its characterization is established.

The class number of an abelian group

The groups in this paper are abelian. Let G be a reduced torsion-free finite rank group. Then G is cocommutative if End ( G ) is commutative modulo the nil radical. The class number of G, h ( G ) , is the number of isomorphism classes of groups H that are locally isomorphic ( = nearly isomorphic) to G. We say that G satisfies the power cancellation property if G n ≅ H n for some group H and integer n > 0 implies that G ≅ H . We say that G has a Σ-unique decomposition if G n has a unique direct sum decomposition for each integer n > 0 . Let P o ( G ) = { groups H | H ⊕ H ′ = G m for some group H ′ and some integer m > 0 } . We say that G has internal cancellation if given H , K , L ∈ P o ( G ) such that H ⊕ K ≅ H ⊕ L then K ≅ L . We use the class number to study the torsion-free finite rank groups G that have the power cancellation property, or a Σ-unique decompositions, or the internal cancellation property. Furthermore, we show that the power cancellation property for cocommutative strongly indecomposable reduced torsion-free finite rank groups is equivalent to the problem of determining the class number of an algebraic number field. Let G ¯ be the integral closure of G. Using the Mayer–Vietoris sequence we show that there are finite groups associated with G of orders L ( p ) , m ˆ p , and n ˆ p such that h ( G ) = h ( G ¯ ) m ˆ p L ( p ) n ˆ p . Let E ( p ) = Z + p E ¯ where p is a rational prime and E ¯ is the ring of algebraic integers in the algebraic number field k = Q E ¯ . Let G ( p ) be a group such that End ( G ( p ) ) = E ( p ) . We show that the sequence { L ( p ) h ( G ( p ) ) / h ( G ¯ ( p ) ) | primes p } is asymptotically equal to the sequence { p f − 1 | primes p } where f = [ k : Q ] . Furthermore, for quadratic number fields k, h ( k ) = 1 iff { L ( p ) h ( G ( p ) ) | primes p } is asymptotically equal to the sequence of rational primes. This connects unique factorization in number fields with the sequence of rational primes, and with direct sum properties of integrally closed cocommutative strongly indecomposable torsion-free finite rank groups.

Jacobson Radicals of Ore Extensions of Derivation Type

Let D be a sequence of derivations of a ring R. We form the Ore extension R[X; D]. Let 𝒩(R) denote the nil radical of R and 𝒥(R[X; D]), the Jacobson radical of R[X; D]. We show that 𝒩(R) = 0 implies 𝒥(R[X; D]) = 0 under one of the following conditions: (1) R is a PI-ring. (2) One of derivations in D is quasi-algebraic. (3) R satisfies the ascending chain condition on right annihilators.

GENERALIZED ENDOPRIMAL ABELIAN GROUPS

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

A radical for right near-rings: The right Jacobson radical of type-0

The notions of a right quasiregular element and right modular right ideal in a near-ring are initiated. Based on theseJ0r(R), the right Jacobson radical of type-0 of a near-ringRis introduced. It is obtained thatJ0ris a radical map andN(R)⊆J0r(R), whereN(R)is the nil radical of a near-ringR. Some characterizations ofJ0r(R) are given and its relation with some of the radicals is also discussed.

Rigid Ideals and Radicals of Ore Extensions

For an endomorphism σ of a ring R, Krempa called σ a rigid endomorphism if aσ(a)=0 implies a=0 for a∈R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the σ-rigid property of a ring R to an ideal of R. For a σ-ideal I of a ring R, we call I a σ-rigid ideal if aσ(a)∈I implies a∈I for a∈R. We characterize σ-rigid ideals and study related properties. The connections of the prime radical and the upper nil radical of R with the prime radical and the upper nil radical of the Ore extension R[x;σ,δ], respectively, are also investigated.

Algebraic radicals and incidence algebras

If I (X, R) denotes the incidence algebra of the locally finite partially ordered set X over the ring with identity R, the upper and lower nil radicals of I (X, R) are described using the strong product property. An application of this information in the determination of algebraic radicals in the case when R is an algebra over a field is then given.

Nil Ideals of Crossed Products

Let ℒ(R) and 𝒦(R) be respectively the locally nilpotent radical and the upper nil radical of the ring R. If K ρ Gis a twisted group ring of a group Gover a ring Kand the order of every torsion element of Gis not a zero divisor in K, then we prove that Moreover, let K ∗ Gbe a crossed product of a group Gover a prime ring Kof characteristic p ≥ 0. If the subgroup G inn has no p-elements when p > 0 and 𝒦(K) = O, then 𝒦(K ∗ G) = O.

The radical in alternative baric algebras

In this paper we proved that in a baric, power associative, F-algebra of finite dimension, there is an idempotent element of weight 1. When this algebra is alternative, with this idempotent we make the Peirce's decomposition. We define the nil radical as the maximal nil ideal. We proved that the bar-radical is the intersection of the nil radical with the square of \(bar({\cal a})\). All the results are naturally true for baric associative algebras of finite dimension.