scholarly journals Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

2016 ◽  
Vol 35 ◽  
pp. 1-5
Author(s):  
Subrata Majumdar ◽  
Kalyan Kumar Dey
Keyword(s):  
Jacobson Radical ◽  
Artinian Rings ◽  
Nil Radical

Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ? N?.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 1-11

scholarly journals A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350121 ◽  
Author(s):  
AGATA SMOKTUNOWICZ
Keyword(s):  
Jacobson Radical ◽  
Analogous Result ◽  
Graded Rings ◽  
Radical Ring ◽  
Graded Ring ◽  
Nil Ring ◽  
Nil Radical

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil radicals, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper, it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained.

Classical Radicals of Invariants of Algebraic Skew Derivations

2022 ◽  
Vol 29 (01) ◽  
pp. 53-66
Author(s):  
Jeffrey Bergen ◽  
Piotr Grzeszczuk
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Nilpotent Radical ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Nil Radical

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].

scholarly journals A radical for right near-rings: The right Jacobson radical of type-0

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad
Keyword(s):  
Jacobson Radical ◽  
The Right ◽  
Nil Radical

The notions of a right quasiregular element and right modular right ideal in a near-ring are initiated. Based on theseJ0r(R), the right Jacobson radical of type-0 of a near-ringRis introduced. It is obtained thatJ0ris a radical map andN(R)⊆J0r(R), whereN(R)is the nil radical of a near-ringR. Some characterizations ofJ0r(R) are given and its relation with some of the radicals is also discussed.

scholarly journals Radical Q-algebras

1976 ◽  
Vol 17 (2) ◽  
pp. 119-126
Author(s):  
P. G. Dixon
Keyword(s):  
Maximal Ideal ◽  
Jacobson Radical ◽  
Uniform Algebra ◽  
Uniform Algebras ◽  
Weighted Sequence ◽  
Sequence Algebra ◽  
Nil Radical

The purpose of this paper is to exhibit various Q-algebras (quotients of uniform algebras) which are Jacobson radical. We begin by noting easy examples of nilpotent Q-algebras and Q-algebras with dense nil radical. Then we describe two ways of constructing semiprime, Jacobson radical Q-algebras. The first is by directly constructing a uniform algebra and an ideal. This produces a nasty Q-algebra as the quotient of a nice uniform algebra (in the sense that it is a maximal ideal of R(X) for some X ⊆ ℂ). The second way is by using results of Craw and Varopoulos to show that certain weighted sequence algebras are Q-algebras. In fact we show that a weighted sequence algebra is Q if the weights satisfy (i) w(n+1)/w(n)↓0 and (ii) (w(n+l)/w(n))∊lp for some p≧ 1, but may be non-Q if either (i) or (ii) fails. This second method produces nice Q-algebras which are quotients of rather horrid uniform algebras as constructed by Craw's Lemma.

Certain Artinian Rings are Noetherian

1972 ◽  
Vol 24 (4) ◽  
pp. 553-556 ◽  
Author(s):  
Robert C. Shock
Keyword(s):  
Jacobson Radical ◽  
Associative Ring ◽  
Identity Element ◽  
Sufficient Conditions ◽  
Artinian Ring ◽  
Composition Series ◽  
Famous Theorem ◽  
Artinian Rings ◽  
Artinian Module ◽  
Necessary And Sufficient

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

The algebra of a commutative semigroup over a commutative ring with unity

1985 ◽  
Vol 99 (3-4) ◽  
pp. 387-398 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Nil Radical

SynopsisA new description is provided for the nil radical of the algebra RS of a commutative semigroup S over a commutative ring R with a 1. It is shown that the Jacobson radical of RS is nil if the Jacobson radical of R is nil and that the converse holds in the case where S is periodic.

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.

scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

Simultaneous Triangularization of Algebras of Polynomially Compact Operators

1991 ◽  
Vol 34 (2) ◽  
pp. 260-264 ◽  
Author(s):  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Compact Operators ◽  
General Algebra ◽  
Closed Algebra ◽  
Quasinilpotent Operators ◽  
Simultaneous Triangularization

AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

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