Jacobson Radical Research Articles

On J-Diagrams for the One Groups of Finite Chain Rings

Let R be a finite commutative chain ring with invariants p,n,r,k,m. The purpose of this article is to study j-diagrams for the one group H=1+J(R) of R, where J(R)=(π) is Jacobson radical of R. In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of {πk′}, where k′∣k. The invariants p,n,r,k,m and the Eisenstein polynomial by which R is constructed over its Galois subring determine fully the j-diagram for H.

Weakly J-submodules of modules over commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a unitary [Formula: see text]-module. By [Formula: see text] we denote the Jacobson radical of [Formula: see text]. The purpose of this paper is to introduce the concept of weakly [Formula: see text]-submodules generalizing [Formula: see text]-submodules. We call a proper submodule [Formula: see text] of [Formula: see text] a weakly [Formula: see text]-submodule if whenever [Formula: see text] for [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Various properties and characterizations of weakly [Formula: see text]-submodules are investigated especially in the case of multiplication modules.

Some characterizations of strongly π-regular rings

A ring R is said to be strongly π-regular if for every aR there exist a positive integers n, bR such that an = an+1b. In this paper it has been proved that an abalian ring R is strongly π-regular if and only the set of all nilpotent element of R coincide with the Jacobson radical and R/J(R) is strongly-regular. In this study, some other characterizations of this kind of rings have been investigated and explored.

On the ideal-based zero-divisor graph of commutative rings

Let [Formula: see text] be a finite commutative ring with identity, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] denotes the Jacobson radical of [Formula: see text]. The ideal-based zero-divisor graph [Formula: see text] of [Formula: see text] is a graph with vertex set [Formula: see text] for some [Formula: see text] in which distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine the diameter, girth of [Formula: see text]. Specifically, we classify all finite commutative nonlocal rings for which [Formula: see text] is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of [Formula: see text] and characterize all of them.

Graded (quasi-)Frobenius rings

In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.

Nearly Endo-T-ABSO submodule and related concepts

In this paper, we discuss the notions Nearly-Endo prime submod, Nearly- EndoT-Absorbing submod, observations, and the traits and characteristics of Endo-T-ABSO and Jacobson radical in relation to one another. The relationship between EndoT-Absorbing submod, Endo prime submod, Nearly-End-prime submod, Nearly-EndoT-Absorbing submods were discovered. We are now introducing submods of several types. The ramifications of this study can be used to create new Nearly-EndoT-Absorbing submod, Nearly-Endo-prime submod based notions.

J-Prime submodules and some related concepts

Suppose R has been an identity-preserving commutative ring, and suppose V has been a legitimate submodule of R-module W. A submodule V has been J-Prime Occasionally as well as occasionally based on what’s needed, it has been acceptable: x ∈ V + J(W) according to some of that r ∈ R, x ∈ W and J(W) an interpretation of the Jacobson radical of W, which x ∈ V or r ∈ [V: W] = {s ∈ R; sW ⊆ V}. To that end, we investigate the notion of J-Prime submodules and characterize some of the attributes of has been classification of submodules.

Generalized Koszul algebra and Koszul duality

We define generalized Koszul modules and rings and develop a generalized Koszul theory for N-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite N-graded ring generated in degree 1 with A0 noetherian semiperfect, J be its graded Jacobson radical. By the Koszul dual of A we mean the Yoneda Ext ring Ext_A•(A/J,A/J). If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is the associated graded ring GrJA and the Koszul dual of the Koszul dual of M is the associated graded module GrJM. If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual Ext_A•(A/J,A/J) of A is (classically) Koszul; GrJA is (classically) Koszul; the opposite ring Aop of A is generalized Koszul. As an application, it is proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective.

О сопряженном к пространству функций, голоморфных на замкнутом полидиске, с мультипликативной сверткой

The algebra of analytic functionals on the closed unit polydisk 𝑫𝟏 ̅̅̅̅ in ℂ𝑵 is studied. The Hadamard type operators are characterized in the space 𝑯(𝑫𝟏 ̅̅̅̅) of all germs of holomorphic on 𝑫𝟏 ̅̅̅̅ functions. We obtain the representation of mentioned algebra in 𝑯(𝑫𝟏 ̅̅̅̅) as the Hadamard type operators algebra. The multiplicative functionals on it, its Jacobson radical and idempotents are described.

On the matrix wreath product of algebras

We study the singular ideal, the Jacobson radical and the maximal quotient rings of the matrix wreath product of two algebras and certain subrings as defined in (Alamhadi-Jain-Alsulami and Zelmanov, Electron. Res. Announc. Math. Sci. (2017)) [1]. We give formulas for computing the singular ideal and the Jacobson radical of a matrix wreath product. We also give the structure of the maximal right quotient ring of the infinite matrix ring of a right nonsingular algebra and, as consequence, we give the maximal quotient ring of a matrix wreath product. As an application of the fact that matrix wreath product of nonsingular algebras is nonsingular we construct another algebra (similar to the one in [3]) showing that any nonsingular algebra of countable dimension can be embedded in a finitely generated nonsingular algebra.

Computing the strong metric dimension for co-maximal ideal graphs of commutative rings

Let [Formula: see text] be a commutative ring with identity. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertices are proper ideals of [Formula: see text] which are not contained in the Jacobson radical [Formula: see text] of [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we use Gallai’s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.

On uniquely clean elements

We introduce a subclass of semipotent rings in which every element which is both uniquely clean and strongly clean can be written as a sum of a central idempotent and an element in the jacobson radical. This, in particular, settles a question left open in [4].

Skew McCoy rings and σ-compatibility

In this paper, we study [Formula: see text]-skew McCoy rings under the [Formula: see text]-compatible or the [Formula: see text]-semicompatible conditions. We show that if [Formula: see text] is a semicommutative right or left artinian ring which is [Formula: see text]-semicompatible with an epimorphism [Formula: see text], then the Jacobson radical [Formula: see text] is [Formula: see text]-skew McCoy. As a corollary, we get that the Jacobson radical of a semicommutative artinian ring is right McCoy. We also show that every [Formula: see text]-compatible right duo ring is [Formula: see text]-skew McCoy and that for [Formula: see text]-compatible regular rings, the notions of the [Formula: see text]-skew McCoy and the right McCoy coincide. In addition, we show that every [Formula: see text]-semicompatible semicommutative ring is linearly [Formula: see text]-skew Camillo and that every matrix ring over a division ring is linearly [Formula: see text]-skew Camillo for any endomorphism [Formula: see text].

On weakly radical ideals of non-commutative rings

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 [Formula: see text] and show that if [Formula: see text] is the Jacobson radical many of the results proved by Khashan et al. are also satisfied for non-commutative rings. The results for weakly J-ideals for non-commutative rings follow as special cases of a more general situation.

Two-sided zero product properties on symmetric algebras

We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ(x,y)=0 whenever xy=yx=0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x,y)↦τ1(xy)+τ2(yx) for x,y∈A, where τ1,τ2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A, there exists a∈A such that, for all x∈A, d(x)−[a,x] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd.

RD-flatness and RD-injectivity of simple modules

We say that a ring R is a right RDV-ring if each simple right R -module is RD -injective. In this paper, we study the notion of RDV -rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD -rings, serial rings and right duo right uniserial rings are RDV -rings. Several characterizations of right RDV -rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J , each simple right R -module is RD -flat if and only if R is a left RDV -ring, if and only if R ( R / J ) is RD -injective, if and only if ( R / J ) R is RD -flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R ( R / J ) is RD -injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R -module is RD -injective (or, R is Köthe), it suffices to test only the RD -injectivity of the simple left R -modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts.

Nearly prime GR-submodules of GR-modules

A gamma ring(Γ− ring) is ΓR-module and W is proper ΓR-submodule. Let W be a proper ΓR-submodule of an ΓR-module is called nearly prime, for short N-P.ΓR-submodule if whenever and rγ m ∈ W implies that either or where is the Jacobson radical of . The goal of this work is to offer an intriguing way of defining and studying nearly P.ΓR-submodule of ΓR-module, for short N-P.ΓR-submodule of ΓR-module result N-P.ΓR-submodule of ΓR-module which represent generalization N-prime submodules.

Ideal spaces: An extension of structure spaces of rings

Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of nontrivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.

On the derivations, generalized derivations and ternary derivations of degree n

In this paper, we introduce the concepts of derivation of degree n, generalized derivation of degree n and ternary derivation of degree n, where n is a positive integer, and then we study the algebraic properties of these mappings. For instance, we study the image of derivations of degree n on algebras and in this regard we prove that, under certain conditions, every derivation of degree n on an algebra maps the algebra into its Jacobson radical. Also, we present some characterizations of these mappings on algebras. For example, under certain assumptions, we show that if f is an additive generalized derivation of degree n with an associated mapping d, then either f is a linear generalized derivation with the associated linear derivation d or f and d are identically zero. Some other related results are also established.

D_j -Supplemented Modules

If \U0001d49c ≪ j Dj (\U0001d4b2), where Dj (\U0001d4b2) is the generalised Jacobson radical of \U0001d4b2, then a sub-module \U0001d49c of a module \U0001d4b2 is said to be Dj -small. If and , \U0001d49c is Dj - ’supplement of ℬ in \U0001d4b2. \U0001d4b2 is said to be Dj - ’supplemented if every sub-module of \U0001d4b2 has Dj - ’supplement in \U0001d4b2. If for each \U0001d49c a sub-module of \U0001d4b2, there exists a sub-module ℬ such that and , then \U0001d4b2 is called Dj -weakly ’supplemented module.