Jacobson Radical Research Articles

Some properties of a graph associated with a non-quasi-local weakly factorial domain and a supergraph of it

Let [Formula: see text] be a weakly factorial domain which admits at least two maximal ideals. We use [Formula: see text] to denote the set of all primary elements of [Formula: see text] and denote [Formula: see text] by [Formula: see text]. Let [Formula: see text]. We denote the subset of [Formula: see text] consisting of all [Formula: see text] such that [Formula: see text] does not belong to the Jacobson radical of [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In Secs. 2 and 3 of this paper, we discuss results regarding the connectedness, the girth, and the clique number of [Formula: see text] and study the interplay between graph properties of [Formula: see text] and the properties of [Formula: see text]. In Secs. 4 and 5 of this paper, we consider a supergraph of [Formula: see text], denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and study some graph properties of [Formula: see text].

Rings with un − 1 nilpotent for each unit u

We continue the study in-depth of the so-called [Formula: see text]-UU rings for any [Formula: see text], that were defined by the first-named author in [Danchev, On exchange P-UU unital rings, Toyama Math. J. 39(1) (2017) 1–7] as those rings [Formula: see text] for which [Formula: see text] is always a nilpotent for every unit [Formula: see text]. Specifically, for any [Formula: see text], we prove that a ring is strongly [Formula: see text]-nil-clean if, and only if, it is simultaneously strongly [Formula: see text]-regular and an [Formula: see text]-UU ring. This somewhat extends results due to [Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211; Abyzov, Strongly q-nil-clean rings, Sib. Math. J. 60(2) (2019) 197–208; Cui and Danchev, Some new characterizations of periodic rings, J. Algebra Appl. 19(12) (2020)]. Moreover, our results somewhat improves the ones obtained by Koşan et al. [Rings such that, for each unit [Formula: see text], [Formula: see text] belongs to the Jacobson radical, Hacettepe J. Math. Stat. 49(4) (2020) 1397–1404].

A A note on the Jacobson radical J ( 𝔽 2 n D 2 m )

UDC 512.5 The dimension of the Jacobson radical J ( 𝔽 2 n D 2 m ) and the algebraic structure of 𝔽 2 n D 2 k 3 J ( 𝔽 2 n D 2 k 3 ) are determined. Here, D 2 m represents the dihedral group of order 2 m , 𝔽 2 n represents any finite field of characteristic 2 , and F 2 n D 2 m represents the group algebra of the group D 2 m over the field 𝔽 2 n .

On the unit-Jacobson graph

In this paper, we introduce the unit-Jacobson graph, which is defined by the unit elements and the elements of the Jacobson radical of a commutative ring [Formula: see text] with nonzero identity. We give relationships between this new graph concept and some special rings such as [Formula: see text]-clean rings, [Formula: see text]-rings, local rings, and cartesian rings. Moreover, we investigate the concepts of the dominating set, diameter, and girth on the unit-Jacobson graph.

On the Jacobson radical of a semiring

Based on the minimal and simple representations of semirings, we introduce two Jacobson-type Hoehnke radicals, namely, m-radical and s-radical, of a semiring [Formula: see text]. Every minimal (simple) [Formula: see text]-semimodule is a quotient of [Formula: see text] by a regular right congruence (maximal) [Formula: see text] on [Formula: see text] such that [Formula: see text] is a maximal [Formula: see text]-saturated right ideal in [Formula: see text]. Thus the m(s)-radical becomes an intersection of a class of regular congruences. The m(s)-radical of the direct product of two semirings is the product of their individual [Formula: see text]-radicals.

Topological Jacobson Radicals. III

Topological Jacobson Radicals. III

One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S

Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has aradical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremumbeing finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2 (P(Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case.

Rings with 2 in the Jacobson radical and generalized bleached conditions

We give an example of a uniquely clean ring R which is not bleached. This example shows that if a ring R is USC then neither R needs to be uniquely totally bleached nor upper triangular matrix ring over R needs to be USC. Also, we give a necessary and sufficient conditions for an upper triangular matrix ring over a ring R to be uniquely strongly clean.

Homological dimensions of the Jacobson radical

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring.

The associative algebra of derivations of a group algebra

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group [Formula: see text] over a finite field [Formula: see text] are determined such that the set of derivations of the group algebra form an associative [Formula: see text]-algebra. The derivations of [Formula: see text] form a nontrivial associative [Formula: see text]-algebra if and only if [Formula: see text] has characteristic 2 and [Formula: see text] is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting [Formula: see text]-algebra is determined.

A Note on the Jacobson Radical of a Graded Ring

We prove that $J(R_e)=R_e\cap J(R),$ where $S$ is a cancellative partial groupoid with idempotent $e,$ $R=\bigoplus_{s\in S}R_s$ an Artinian $S$-graded ring inducing $S,$ $J(R)$ the Jacobson radical of $R$ and $J(R_e)$ the Jacobson radical of $R_e.$ We also prove that $J(R)$ is nil if $J(R_e)$ is nil under certain assumptions.

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.

About the image of strongly generalized derivations of order $n$

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalizedderivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐹_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐺_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴.The main purpose of this paper is to study the image of such derivations. Our main result onthe image of strongly generalized derivations of order one reads as follows: Let A be a unital,commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalizedderivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linearmappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained inthe Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image ofcontinuous derivations on commutative Banach algebras.

A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R [ x ; α , δ ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

A Note on Weakly Semiprime Ideals and Their Relationship to Prime Radical in Noncommutative Rings

In this paper, we introduce the concept of weakly semiprime ideals and weakly n-systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R−P being a weakly n-system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide.

When every S-flat module is (flat) projective

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS , is a flat RS -module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the “S-perfectness” for various ring constructions, which allows the construction of more interesting examples.

When unit graphs are isomorphic to unitary Cayley graphs of rings?

Let [Formula: see text] be a ring with identity. The unit (respectively, unitary Cayley) graph of [Formula: see text] is a simple graph [Formula: see text] (respectively, [Formula: see text]) with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] (respectively, [Formula: see text]) is a unit of [Formula: see text]. In this paper, we explore when [Formula: see text] and [Formula: see text] are isomorphic for a finite ring [Formula: see text]. Among other results, we prove that [Formula: see text] is isomorphic to [Formula: see text] for a finite ring [Formula: see text] if and only if char([Formula: see text])[Formula: see text]=[Formula: see text]2 or [Formula: see text], where [Formula: see text] is the Jacobson radical of [Formula: see text] and [Formula: see text] is a finite ring.

Irreducible graded bimodules over algebras and a Pierce decomposition of the Jacobson radical

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char ( F ) = 0 , A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A = M n ( F σ [ H ] ) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and σ ∈ Z 2 ( H , F * ) , then M being irreducible graded implies that there exists a nonzero homogeneous element w ∈ M satisfying M = B w and B w = w B . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w 1 , w 2 , … , w n ∈ M such that M = A w 1 ⊕ A w 2 ⊕ ⋯ ⊕ A w n , where w i A = A w i ≠ 0 for all i = 1 , 2 , … , n , and each A w i is irreducible. The elements wi ’s are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading.

Rings with u − 1 quasinilpotent for each unit u

In this paper, we define and explore in-depth the notion of UQ rings by showing their important properties and by comparing their behavior with that of the well-known classes of UU rings and JU rings, respectively. Specifically, among the other established results, we prove that UQ rings are always Dedekind finite (often named directly finite) as well as that, for semipotent rings [Formula: see text], the following equivalence hold: [Formula: see text] is [Formula: see text] is UJ having the property that the set [Formula: see text] of quasinilpotent elements of [Formula: see text] coincides with the Jacobson radical [Formula: see text] of [Formula: see text].

Jacobson radicals, abelian $p$-groups and the $\oplus_{c}$-topology

In studying the Jacobson radical of the endomorphism ring of a separable abelian p -group, Sands (1984) identified the useful Condition (C). Our central result is that the group G satisfies Condition (C) precisely when it is complete in its \oplus_{c} -topology , which uses the set of subgroups X\leq G such that G/X is a direct sum of cyclics as a neighborhood base of 0. This equivalence is then used to compute the Jacobson radicals of the endomorphism rings of a variety of such groups, including those in the so-called Keef class . It is shown that Sands’ attempt to “complete” an arbitrary group with respect to Condition (C) is equivalent to D’Este’s (1980) attempt to show that the \oplus_{c} -topology is completable. Since Mader (1983) provided a counterexample to D’Este’s result, it follows that Sands’ result also fails.