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 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.

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.

scholarly journals On p-injective rings

1995 ◽  
Vol 37 (3) ◽  
pp. 373-378 ◽  
Author(s):  
Gennadi Puninski ◽  
Robert Wisbauer ◽  
Mohamed Yousif
Keyword(s):  
Direct Summand ◽  
Jacobson Radical ◽  
Associative Ring ◽  
The Right ◽  
Left Annihilator

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

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].

ON SMALL INJECTIVE RINGS AND MODULES

2009 ◽  
Vol 08 (03) ◽  
pp. 379-387 ◽  
Author(s):  
LE VAN THUYET ◽  
TRUONG CONG QUYNH
Keyword(s):  
Jacobson Radical ◽  
Injective Module ◽  
The Right

A right R-module MR is called small injective if every homomorphism from a small right ideal to MR can be extended to an R-homomorphism from RR to MR. A ring R is called right small injective, if the right R-module RR is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module ER, E(ℕ) is small injective.

scholarly journals Right alternative algebras with commutators in a nucleus

1992 ◽  
Vol 46 (1) ◽  
pp. 81-90
Author(s):  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Associative Ring ◽  
Linear Span ◽  
Alternative Algebra ◽  
Wedderburn Decomposition ◽  
Alternative Algebras ◽  
Finite Dimensional ◽  
Characteristic 2 ◽  
The Right ◽  
Nil Radical ◽  
Algebra Over A Field

Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

scholarly journals Coincidence of topological Jacobson radicals in topological algebras

2017 ◽  
Vol 21 (2) ◽  
pp. 239-247
Author(s):  
Mart Abel ◽  
Mati Abel ◽  
Paul Tammo
Keyword(s):  
Jacobson Radical ◽  
Topological Algebras ◽  
The Right

Several classes of topological algebras for which the left topological Jacobson radical coincides with the right topological Jacobson radical are described.

scholarly journals ON 𝒯-NONCOSINGULAR MODULES

2009 ◽  
Vol 80 (3) ◽  
pp. 462-471 ◽  
Author(s):  
DERYA KESKIN TÜTÜNCÜ ◽  
RACHID TRIBAK
Keyword(s):  
Jacobson Radical ◽  
The Right

AbstractIn this paper we introduce 𝒯-noncosingular modules. Rings for which all right modules are𝒯-noncosingular are shown to be precisely those for which every simple right module is injective. Moreover, for any ring R we show that the right R-module R is 𝒯-noncosingular precisely when R has zero Jacobson radical. We also study the 𝒯-noncosingular condition in association with (strongly) FI-lifting modules.

scholarly journals On the radical of a group algebra over a commutative ring

1977 ◽  
Vol 18 (1) ◽  
pp. 101-104 ◽  
Author(s):  
Gloria Potter
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Algebra ◽  
Jacobson Radical ◽  
Multiplicative Group ◽  
Commutative Rings ◽  
The Other ◽  
Group Algebras ◽  
Study Group ◽  
The Right

Several people, including Wallace [4] and Passman [3], have studied the Jacobson radical of the group algebra F[G] where F is a field and G is a multiplicative group. In [4], for instance, Wallace proves that if G is an abelian group with Sylow p-subgroup P and if F is a field of characteristic p, then the Jacobson radical of F[G] equals the right ideal generated by the radical of F[P]. In this paper we shall study group algebras over arbitrary commutative rings. By a reduction to the case of a semi-simple commutative ring, we obtain Theorem 1 whose corollary contains a generalization of Wallace's theorem. Theorem 2, on the other hand, uses the first theorem to obtain results related to the main theorem of [3].

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.

Sign in / Sign up

Export Citation Format

Share Document