Jacobson Radical Research Articles

Banach algebras satisfying certain chain conditions on closed ideals

AbstractLet 𝔄 be a unital Banach algebra and ℜ its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra 𝔄 satisfies the descending chain condition on closed left ideals then 𝔄/ℜ is finite dimensional. We also prove that a C*-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

The jacobson radical types of Leavitt path algebras with coefficients in a commutative unital semiring

In this paper, we calculate the J - radical and Js - radical of the Leavitt path algebras with coefficients in a commutative semiring of some finite graphs. In particular, we calculate J - radical and Js - radical of the Leavitt path algebras with coefficients in a field of acyclic graphs, no-exit graphs and give applicable examples.

Star-fundamental algebras: polynomial identities and asymptotics

We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution ∗ * . To any star-algebra A A is attached a numerical sequence c n ∗ ( A ) c_n^*(A) , n ≥ 1 n\ge 1 , called the sequence of ∗ * -codimensions of A A . Its asymptotic is an invariant giving a measure of the ∗ * -polynomial identities satisfied by A A . It is well known that for a PI-algebra such a sequence is exponentially bounded and exp ∗ ⁡ ( A ) = lim n → ∞ c n ∗ ( A ) n \exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)} can be explicitly computed. Here we prove that if A A is a star-fundamental algebra, C 1 n t exp ∗ ⁡ ( A ) n ≤ c n ∗ ( A ) ≤ C 2 n t exp ∗ ⁡ ( A ) n , \begin{equation*} C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, \end{equation*} where C 1 > 0 , C 2 , t C_1>0,C_2, t are constants and t t is explicitly computed as a linear function of the dimension of the skew semisimple part of A A and the nilpotency index of the Jacobson radical of A A . We also prove that any finite dimensional star-algebra has the same ∗ * -identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144–167] we get that if A A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, lim n → ∞ log n ⁡ c n ∗ ( A ) exp ∗ ⁡ ( A ) n \lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n} exists and is an integer or half an integer.

A generalization of UJ-rings

The aim of the paper is to characterize the equation [Formula: see text], where [Formula: see text] that is the largest Jacobson radical subring of [Formula: see text] and [Formula: see text] is the set of invertible elements of a ring [Formula: see text]. We show that this equation is closely related to [Formula: see text]-rings and rings whose elements can be written as the sum of an idempotent and an element from [Formula: see text]. After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation [Formula: see text] within many well-studied classes of rings. Finally, we close the paper with group rings.

Pseudo core inverses of a sum of morphisms

Let be an additive category with an involution ∗. Suppose that both ϕ : X → X is a morphism of with pseudo core inverse ϕ and η : X → X is a morphism of such that 1 + ϕ η is invertible. Let α = (1 + ϕ η) − 1 , β = (1+ηϕ ) − 1 , ε = (1−ϕϕ )ηα(1−ϕ ϕ), γ = α(1−ϕ ϕ)ηϕ β, σ = αϕ ϕα− 1(1− ϕϕ )β, δ = β∗ (ϕ ) ∗η∗ (1−ϕϕ )β. Then we present a sufficient condition such that f = ϕ + η − ε has pseudo core inverse and give the corresponding expression. Let R be a unital ∗-ring and J(R) its Jacobson radical. If a is pseudo core invertible with the pseudo core inverse a and j ∈ J(R), we also give a sufficient condition which ensures that a + j − ε has pseudo core inverse. Thus, these results generalize recent results on core inverse.

Rings such that, for each unit u, u − u n belongs to the Jacobson radical

A ring R is said to be n-UJ if u − u n ∈ J(R) for each unit u of R, where n > 1 is a fixed integer. In this paper, the structure of n-UJ rings is studied under various conditions. Moreover, the n-UJ property is studied under some algebraic constructions. Mathematics Subject Classification (2010). 16N20, 16D60, 16U60, 16W10

On the primitive ideals of nest algebras

AbstractWe show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

On the spectrum of the closed unit graphs

Let R be a finite commutative ring with non-zero identity. Let and be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by , is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if . If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit graph, denoted by . In this paper, we compute the adjacency spectrum of the graph . We utilize this result to show that if and only if and , where R and S are two arbitrary finite rings. Moreover, we determine when is a Ramanujan graph. We also deliver a necessary and sufficient condition for to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs.

Co-maximal Graph, its Planarity and Domination Number

Let R be a finite commutative ring with identity. The co-maximal graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also, Γ2(R) is the subgraph of Γ(R) induced by non-unit elements and [Formula: see text] where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which [Formula: see text], Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.

The Jacobson Radical of the Endomorphism Semiring of P.Q.- Principal Injective Semimodules


 In this work, we introduced the Jacobson radical (shortly Rad (Ș)) of the endomorphism semiring Ș = ( ), provided that is principal P.Q.- injective semimodule and some related concepts, we studied some properties and added conditions that we needed. The most prominent result is obtained in section three
 -If is a principal self-generator semimodule, then (ȘȘ) = W(Ș).
 Subject Classification: 16y60

Strongly P-Clean and Semi-Boolean Group Rings

A ring $R$ is called clean (resp., uniquely clean) if every element is (uniquely represented as) the sum of an idempotent and a unit. A ring $R$ is called strongly P-clean if every its element can be written as the sum of an idempotent and a strongly nilpotent element that commute. The class of strongly P-clean rings is a subclass of classes of semi-Boolean and strongly nil clean rings. A ring $R$ is called semi-Boolean if $R/J(R)$ is Boolean and idempotents lift modulo $J(R),$ where $J(R)$ denotes the Jacobson radical of $R.$ The class of semi-Boolean rings lies strictly between the classes of uniquely clean and clean rings. We obtain a complete characterization of strongly P-clean group rings. It is proved that the group ring $RG$ is strongly P-clean if and only if $R$ is strongly P-clean and $G$ is a locally finite 2-group. Further, we also study semi-Boolean group rings. It is proved that if a group ring $RG$ is semi-Boolean, then $R$ is a semi-Boolean ring and $G$ is a 2-group and that the converse assertion is true if $G$ is locally finite and solvable, or an FC group.

Locally nilpotent skew extensions of rings

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if G={σt}t∈T is a locally finite family of automorphisms of an algebra R, D={δt}t∈T is a family of skew derivations of R such that the prime radical P of R is strongly invariant under D, then the ideal P〈T,G,D〉⁎ of R〈T,G,D〉, generated by P, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra R over a field of characteristic 0, having a surjective locally nilpotent derivation d with commutative kernel, and such that R is generated by ker⁡d2, has a locally nilpotent Jacobson radical.

Jacobson graph construction of ring , for n>1

A group is called a ring if the group is a commutative under addition operation and satisfy the distributive and assosiative properties under multiplication operation. Suppose R is a commutative ring with non-zero identity, U is the unit of R, and J(R) is a Jacobson radical. Jacobson graph of a ring R denoted by ℑ R is a graph with a vertex set is R\\J(R) dan edge set is {(a, b)| 1 − ab ∉ U}. The purpose of this research is to construct a Jacobson graph of ring Z 3 n with n > 1. The results show that Jacobson graph of ring Z 3 n is a disconected graph with two components.

The Unipotent Condition of Linear Groups Generated by Matrices with Primitive Elements Jorden Blocks of Orders no More Than Eight

In this paper, the combinatorial properties of free group generators are studied with help of calculation tool, matrix logarithm, and Jacobson radical. The unipotency of matrix groups can be better delineated by these new combinatorial properties, and a new necessary and sufficient unipotent condition of matrix group whose primitive elements Jordan blocks have orders no more than eight, as well as of free groups, will be given. Our result improves the relative theories.

Notes on the graded Jacobson radical: a graded version of the Jacobson stable set

Let R be an S-graded ring inducing S, that is, a ring which is the direct sum of a family of its additive subgroups indexed by a nonempty set S, under the assumption that the product of homogeneous elements is again homogeneous. We introduce a graded version of the subring and discuss its homogeneity, where U(R) denotes the group of units of R.Communicated by Pavel Kolesnikov

On the injective dimension of the Jacobson radical

We conjecture that the injective dimension of the Jacobson radical equals the global dimension for Artin algebras. We provide a proof of this conjecture in case the Artin algebra has finite global dimension and in some other cases.

A new proof of Singer-Wermer Theorem with some results on {g, h}-derivations.

Singer and Wermer proved that if A is a commutative Banach algebra and d: A → A is a continuous derivation, then d(A) ⊆ rad(A), where rad(A) denotes the Jacobson radical of A. In this paper, we establish a new proof of that theorem. Moreover, we prove that every continuous Jordan derivation on a finite dimensional Banach algebra, under certain conditions, is identically zero. As another objective of this article, we study {g, h}-derivations on algebras. In this regard, we prove that if f is a {g, h}-derivation on a unital algebra, then f, g and h are generalized derivations. Additionally, we achieve some results concerning the automatic continuity of {g, h}-derivations on Banach algebras. In the last section of the article, we introduce the concept of a {g, h}-homomorphism and then we present a characterization of it under certain conditions.

A new characterization of commutative semiregular rings

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

On a common generalization of symmetric rings and quasi duo rings

Let \(J(R)\) denote the Jacobson radical of a ring \(R\). We call a ring \(R\) as \(J\)-symmetric if for any \(a,b, c\in R, abc=0\) implies \(bac\in J(R)\). It turns out that \(J\)-symmetric rings are a common generalization of left (right) quasi-duo rings and generalized weakly symmetric rings. Various properties of these rings are established and some results on exchange rings and the regularity of left SF-rings are generalized.

On the lattice of z-ideals of a commutative ring

We prove that the lattice of z-ideals of a commutative ring with identity is a coherent frame. We characterize when it is a Yosida frame, and when it satisfies some projectability properties. We also characterize Hilbert rings in terms of ideals that arise naturally in this study. A ring with zero Jacobson radical is shown to be feebly Baer precisely when its frame of z-ideals is feebly projectable. Denote by ZId(A) the frame of z-ideals of a ring A. We show that the assignment A↦ZId(A) is the object part of a functor CRngz→CohFrm, where CRngz designates the category whose objects are commutative rings with identity and whose morphisms are the ring homomorphisms that contract z-ideals to z-ideals.