Jacobson Radical Research Articles

When is a ϕ-derivation continuous and where can its image be found?

This paper has two main objectives. One of them is to obtain some conditions under which the image of a [Formula: see text]-derivation on a Banach algebra [Formula: see text] is contained in the Jacobson radical of [Formula: see text] and other purpose of this study is to present some results about the automatic continuity of [Formula: see text]-derivations in Banach algebras.

Symmetry on zero and idempotents

Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, for implies and for implies respectively, where S is a subset of a ring R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial over a symmetric ring is a unit then a 0 is a unit and ai is nilpotent for all based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.

Some results on Gorenstein (weak) global dimension of rings

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. We give support to the principle. We prove that the Gorenstein weak global dimension of a ring R equals the supremum of the Gorenstein flat dimensions of the finitely presented cyclic (left or right) R-modules. We also show that, if R is a coherent ring and x is a central nonzero-divisor in R contained in the Jacobson radical of R, then Finally, we give the Gorenstein counterpart of the well-known Hilbert’s Syzygy Theorem.

On the Jacobson radical of a ring

On the Jacobson radical of a ring

Relative projectivity and a property of Jacobson radical

Relative projectivity and a property of Jacobson radical

A note on the Jacobson radical of a hemiring

A note on the Jacobson radical of a hemiring

Domain property of rings relative to the Jacobson radical

Domain property of rings relative to the Jacobson radical

On rings which are sums of subrings and additive subgroups

We prove that if a ring [Formula: see text] is a sum of its Jacobson radical subring [Formula: see text] and additive subgroup [Formula: see text] with [Formula: see text], then the ring [Formula: see text] is Jacobson radical, which answers a question asked by Kelarev and McConnell. Using this result we also show that for a semigroup [Formula: see text] the class of Jacobson radical rings is [Formula: see text]-sum closed if and only if [Formula: see text] is locally finite and all subgroups of [Formula: see text] are [Formula: see text]-groups.

Quasi J-ideals of commutative rings

Let R be a commutative ring with identity. In this paper, we introduce the concept of quasi J-ideal which is a generalization of J-ideal. A proper ideal of R is called a quasi J-ideal if its radical is a J-ideal. Many characterizations of quasi J-ideals in some special rings are obtained. We characterize rings in which every proper ideal is quasi J-ideal. Further, as a generalization of presimplifiable rings, we define the notion of quasi presimplifiable rings. We call a ring R a quasi presimplifiable ring if whenever \(a,b\in R\) and \(a=ab\), then either a is a nilpotent or b is a unit. It is shown that a proper ideal I that is contained in the Jacobson radical is a quasi J-ideal (resp. J-ideal) if and only if R/I is a quasi presimplifiable (resp. presimplifiable) ring.

On radical ideals of non-commutative rings

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. Tekir, S. Koc and K. H. Oral, [Formula: see text]-Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an [Formula: see text] ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define [Formula: see text]-ideals for a special radical [Formula: see text] and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case.

The Jacobson radical for an inconsistency predicate

As a form of the Axiom of Choice about relatively simple structures (posets), Hausdorff’s Maximal Chain Principle appears to be little amenable to computational interpretation. This received view, however, requires revision: maximal chains are more reminiscent of maximal ideals than it seems at first glance. The latter live in richer algebraic structures (rings), and thus are readier to be put under computational scrutiny. Exploiting this, and of course the analogy between maximal chains and maximal ideals, the concept of Jacobson radical carries over from a ring to an arbitrary set with an abstract inconsistency predicate: that is, a distinguished monotone family of finite subsets. All this makes possible not only to generalise Hausdorff’s principle, but also to express it as a syntactical conservation theorem. The latter, which encompasses the desired computational core of Hausdorff’s principle, is obtained by a generalised inductive definition. The over-all setting is constructive set theory.

A Bezout ring with nonzero principal Jacobson radical

In this paper, we study a commutative Bezout domain with nonzero Jacobson radical being a principal ideal. It has been proved that such a Bezout domain is a ring of the stable range 1. As a result, we have obtained that such a Bezout domain is a ring over which any matrix can be reduced to a canonical diagonal form by means of elementary transformations of its rows and columns.

Superalgebras: Polynomial identities and asymptotics

To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞⁡cnsup(A)n is an integer that can be explicitly computed.Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then(1)C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group.As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that limn→∞⁡logn⁡cnsup(A)expsup(A)n exists and is a half integer.

The matrix Jacobson graph of finite commutative rings

The notion of the matrix Jacobson graph was introduced in 2019. Let R be a commutative ring and J ( R ) be the Jacobson radical of ring R . The matrix Jacobson graph of ring R size m × n , denoted 𝔍( R ) m × n , is defined as a graph where the vertex set is R m × n ∖ J ( R ) m × n such that two distinct vertices A , B are adjacent if and only if 1 − det( A t B ) is not a unit in ring R . Here we obtain some graph theoretical properties of 𝔍( R ) m × n including its connectivity, planarity and perfectness.

Decompositions of matrices into potent and square-zero matrices

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

On I-finite left quasi-duo rings

A ring is called left quasi-duo (left QD) if every maximal left ideal is a right ideal, and it is called I-finite if it contains no infinite orthogonal set of idempotents. It is shown that a ring is I-finite and left QD if and only if it is a generalized upper-triangular matrix ring with all diagonal rings being division rings except the lower one, which is either a division ring or it is I-finite, left QD and left `soclin' (left QDS). Here a ring is called left soclin if each simple left ideal is nilpotent. The left QDS rings are shown to be finite direct products of indecomposable left QDS rings, in each of which 1=f1+⋯+fm1=f1+⋯+fm where the fifi are orthogonal primitive idempotents, with fk≈flfk≈fl for all k,l,k,l, and ≈≈ is the block equivalence on {f1,…,fm}.{f1,…,fm}.A ring is shown to be left soclin if and only if every maximal left ideal is left essential, if and only if the left socle is contained in the left singular ideal. These left soclin rings are proved to be a Morita invariant class; and if a ring is semilocal and non-semisimple, then it is left soclin if and only if the Jacobson radical is essential as a left ideal..Left quasi-duo elements are defined for any ring and shown to constitute a subring containing the centre and the Jacobson radical of the ring. The `width' of any left QD ring is defined and applied to characterize the semilocal left QD rings, and to clarify the semiperfect case..

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 the restricted graded Jacobson radical of rings of Morita context

The class of rings $\mathcal{J}=\{A (A,\circ)$ forms a group$\}$ forms a radical class and it is called the Jacobson radical class. For any ring $A$, the Jacobson radical $\mathcal{J}(A)$ of $A$ is defined as the largest ideal of $A$ which belongs to $\mathcal{J}$. In fact, the Jacobson radical is one of the most important radical classes since it is used widely in another branch of abstract algebra, for example, to construct a two-sided brace. On the other hand, for every ring of Morita context $T=\begin{pmatrix} R & V \\ W & S \end{pmatrix}$, we will show directly by the structure of the Jacobson radical of rings that the Jacobson radical $\mathcal{J}(T)= \begin{pmatrix} \mathcal{J}(R) & V_0\\ W_0 & \mathcal{J}(S) \end{pmatrix}$, where $\mathcal{J}(R)$ and $\mathcal{J}(S)$ are the Jacobson radicals of $R$ and $S$, respectively, $V_0=\{v \in V vW \subseteq \mathcal{J}(R)\}$ and $W_0=\{w \in W wV \subseteq \mathcal{J}(S)\}$. This clearly shows that the Jacobson radical is an $N-$radical. Furthermore, we show that this property is also valid for the restricted $G-$graded Jacobson radical of graded ring of Morita context.

Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order pk, k = 2, 3, 4

Let R0 = GR(pkr, pk) be a Galois maximal subring of R so that R = R0 ⊕ U ⊕ V ⊕ W ⊕ Y, where U, V, W, and Y are R0/pR0 spaces considered as R0‐modules, generated by the sets {u1, ⋯, ue}, {v1, ⋯, vf}, {w1, ⋯, wg}, and {y1, ⋯, yh}, respectively. Then, R is a completely primary finite ring with a Jacobson radical Z(R) such that (Z(R))5 = (0) and (Z(R))4 ≠ (0). The residue field R/Z(R) is a finite field GF(pr) for some prime p and positive integer r. The characteristic of R is pk, where k is an integer such that 1 ≤ k ≤ 5. In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with pψui = 0, ψ = 2, 3, 4; pζvj = 0, ζ = 2, 3; pwk = 0, and pyl = 0; 1 ≤ i ≤ e, 1 ≤ j ≤ f, 1 ≤ k ≤ g, and 1 ≤ l ≤ h.

Assessment of Structural Cracks in Aircraft Using a Decision-Making Approach Based on Enhanced Entropy and Single-Valued Neutrosophic Sets

Structural health monitoring (SHM) tries to determine the status of substances, elements, and the life expectancy of any aircraft building. It also serves as a vital tool for ensuring the structural integrity and safety of the aircraft. This research used a neutrosophic-Entropy model to assess the structural cracks in the reinforced concrete aircraft to evaluate its structural performance. For example, Entropy may be used in the civil engineering profession to pick subcontractors, manage traffic, and monitor the structural integrity of a structure, among other things. SHM in reinforced concrete aircraft buildings is examined in this research using the suggested technology. Expert groups were consulted to help determine the level of significance among the parameters used to monitor the aircraft's structural integrity. To further understand the structures' actual improvement, the Entropy approach given here would be quite beneficial. Also, we presented some neutrosophic multiplications. Neutrosophic multiplication module (NMM) (a.k.a. ) (commulative group (A))has produced some surprising new findings, which we explore in this article. As a bonus, we'll shed some light on a few related topics to the NMM itself. Since the Neutrosophic Jacobson radical of the A multiplication modules is a Neutrosophic small submodule of , then must be a Neutrosophic cyclic module, as we demonstrate. Last but not least, we prove that E (Universal) is a NMM only if and only when it's Neutrosophic divided modules more than a Neutrosophic integral domain.