• 한국지능시스템학회논문지
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• 제17권3호
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• pp.423-429
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• 2007

We define and study the fuzzy Jacobson radical of a ${\kappa}$-semiring. Also it is shown that the Jacobson radical of the quotient semiring R/FJR(R) of a ${\kappa}$-semiring by the fuzzy Jacobson radical FJR(R) is semisimple. And the algebraic properties of the fuzzy ideals FJR(R) and FJR(S) under a homomorphism from R onto S are also discussed.

• Korean Journal of Mathematics
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• 제18권1호
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• pp.97-103
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• 2010

It is well known that the group ring K[G] has the nontrivial Jacobson radical if K is a field of characteristic p and G is a finite group of which order is divided by a prime p. This paper is concerned with the structure of the Jacobson radical of such a group ring.

• 대한수학회지
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• 제38권6호
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• pp.1117-1123
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• 2001

A ring is a DICC ring if every chain of right ideals in-dexed by the integers stabilizes to the left or to the right or to both sides. A counterexample is given to an assertion of karamzadeh and Motamedi that a transfinite power of the Jacobson radical of a right DICC ring is zero. we determine the behavior of the transfinite powers of the Jacobson radical relative to a torsion theory and consequently can obtain their correct behavior in the classical setting.

• 대한수학회보
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• 제41권4호
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• pp.599-608
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• 2004

Let R be an associative ring with identity and J(R) be the Jacobson radical of R. In this paper we investigate the generalization of the Jacobson radical of R, J＊ (R) say. Also we study the rings that J＊(R) = J(R).

• 대한수학회논문집
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• 제15권3호
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• pp.453-467
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• 2000

For any S-flat module RM(which will be called endoflat) with a commutaitve ring R with identity, where S is the endomorphism ring RM, the fact that every epimorphism is an automorphism has been proved and the Jacobson Radical Rad(S) of S is described as follow; Rad(S) = { f$\in$S｜Imf=Mf is small in M} = {f$\in$S｜Imf $\leq$Rad(M)}. Additionally for any quasi-injective endo-flat module RM, the fact that every monomorphism is an automorphism has been proved and the Jacobson Radical Rad(S) for any quasi-injective endo-flat module has been studied too. Also some equivalent conditions for the semi-primitivity of any faithful endo-flat module RM with the open Jacobson Radical Rad(M) and those for the semi-simplicity of any faithful endo-flat quasi-injective module RM with the closed Socle Soc(M) have been studied.

• Korean Journal of Mathematics
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• 제21권1호
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• pp.23-28
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• 2013

The structures of finite local rings of order ${\leq}$ 16 with nonzero Jacobson radical are investigated. The whole shape of non-commutative local rings of minimal order is completely determined up to isomorphism.

• Kyungpook Mathematical Journal
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• 제57권1호
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• pp.125-132
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• 2017

We obtain commutativity-free characterizations of higher derivations on a unital Banach algebra that map into its Jacobson radical. We also investigate the spectral boundedness of generalized higher derivations.

• 대한수학회지
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• 제53권2호
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• pp.415-431
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• 2016

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

• 대한수학회보
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• 제37권2호
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• pp.265-272
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• 2000

Let D be a derivation on an Banach algebra A. Suppose that [[D(x), x], D(x)] lies in the nil radical of A for all $x{\;}{\in}{\;}A$. Then D(A) is contained in the Jacobson radical of A.

• East Asian mathematical journal
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• 제34권1호
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• pp.39-46
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• 2018

In this article we consider rings whose Jacobson radical contains all the nilpotent elements, and call such a ring an NJ-ring. The class of NJ-rings contains NI-rings and one-sided quasi-duo rings. We also prove that the Koethe conjecture holds if and only if the polynomial ring R[x] is NJ for every NI-ring R.