scholarly journals A characterization of the Jacobson radical

1963 ◽  
Vol 14 (4) ◽  
pp. 595-595 ◽  
Author(s):  
A. Kert{ész
Keyword(s):  
Jacobson Radical

Spectral characterization of the radical in Banach and Jordan–Banach algebras

1993 ◽  
Vol 114 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Representation Theory ◽  
Spectral Radius ◽  
Jacobson Radical ◽  
Classical Result ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
General Situation

If a is a n × n matrix such that a + m is invertible for every invertible a + m matrix m, then a = 0, by a classical result of Perlis [8]. Unfortunately the same result is not true in general for semi-simple rings as shown by T. Laffey. In the general situation of Banach algebras, Zemánek[12] has proved that a is in the Jacobson radical of A if and only if ρ(a+x) = ρ(x), for every x in A, where ρ denotes the spectral radius. More sophisticated characterizations of the radical were given in [4] and [3], theorem 5·3·1. The arguments used in all these situations depend heavily on representation theory.

q-Hypercyclic Rings

1985 ◽  
Vol 37 (3) ◽  
pp. 452-466 ◽  
Author(s):  
S. K. Jain ◽  
D. S. Malik
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Injective Hull

A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.

scholarly journals On some constructions of nil-clean, clean and exchange rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550101
Author(s):  
Alin Stancu
Keyword(s):  
Jacobson Radical ◽  
Exchange Ring ◽  
Common Theme ◽  
Infinitesimal Deformation ◽  
Formal Deformation ◽  
Exchange Rings ◽  
Infinitesimal Deformations ◽  
The Common

In this paper, we discuss several constructions that lead to new examples of nil-clean, clean and exchange rings. Extensions by ideals contained in the Jacobson radical is the common theme of these constructions. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations, we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean and exchange rings, arising from poset algebras are also discussed.

scholarly journals A characterization of artinian rings

1988 ◽  
Vol 30 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Dinh van Huynh ◽  
Nguyen V. Dung
Keyword(s):  
Jacobson Radical ◽  
Ring Theory ◽  
The Other ◽  
Simple Modules ◽  
Artinian Rings ◽  
Associative Rings

Throughout this paper we consider associative rings with identity and assume that all modules are unitary. As is well known, cyclic modules play an important role in ring theory. Many nice properties of rings can be characterized by their cyclic modules, even by their simple modules. See, for example, [2], [3], [6], [7], [13], [14], [15], [16], [18], [21]. One of the most important results in this direction is the result of Osofsky [14, Theorem] which says: a ring R is semisimple (i.e. right artinian with zero Jacobson radical) if and only if every cyclic right R-module is injective. The other one is due to Vamos [18]: a ring R is right artinian if and only if every cyclic right R-module is finitely embedded.

On the radical of semigroup algebras satisfying polynomial identities

1986 ◽  
Vol 99 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Commutative Semigroup ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Polynomial Identities ◽  
Arbitrary Semigroup

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

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