• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.1
• /
• pp.149-161
• /
• 2010

We adopt the idea of $C{\breve{a}}dariu$ and Radu to prove the generalized Hyers-Ulam stability of generalized Jordan triple derivation in Banach algebra. In addition, we take account of problems for generalized Jordan triple linear derivation in Banach algebra.

• Kyungpook Mathematical Journal
• /
• v.45 no.1
• /
• pp.21-31
• /
• 2005

We introduce the notions of nilpotent element, quasi-regular element in a semiring which is a distributive lattice of rings. The concept of Jacobson radical is introduced for this kind of semirings.

• Kyungpook Mathematical Journal
• /
• v.49 no.1
• /
• pp.67-79
• /
• 2009

We establish the generalized Hyers-Ulam-Rassias stability of higher ring derivations. Furthermore, we use the superstability of higher ring derivations to obtain approximately higher linear derivations mapping into the Jacobson radical under some conditions.

• Journal of the Chungcheong Mathematical Society
• /
• v.18 no.1
• /
• pp.87-93
• /
• 2005

For a left R-module M, we identify certain submodules of M that play a role analogous to that of ideals in the ring R. We investigate some properties of M-ideals in the submodules of M and also study Jacobson radicals of a submodule of M.

• Kyungpook Mathematical Journal
• /
• v.56 no.1
• /
• pp.83-94
• /
• 2016

Let M and N be modules over a ring R. The purpose of this paper is to study modules M, N for which the bi-module [M, N] is regular or pi. It is proved that the bi-module [M, N] is regular if and only if a module N is semi M-projective and $Im({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$, if and only if a module M is semi N-injective and $Ker({\alpha}){\subseteq}^{\oplus}N$ for all ${\alpha}{\in}[M,N]$. Also, it is proved that the bi-module [M, N] is pi if and only if a module N is direct M-projective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Im({\alpha}{\beta}){\subseteq}^{\oplus}N$, if and only if a module M is direct N-injective and for any ${\alpha}{\in}[M,N]$ there exists ${\beta}{\in}[M,N]$ such that $Ker({\beta}{\alpha}){\subseteq}^{\oplus}M$. The relationship between the Jacobson radical and the (co)singular ideal of [M, N] is described.

• Journal of the Korean Mathematical Society
• /
• v.56 no.3
• /
• pp.723-738
• /
• 2019

A property of reduced rings is proved in relation with centers, and our argument in this article is spread out based on this. It is also proved that the Wedderburn radical coincides with the set of all nilpotents in symmetric-over-center rings, implying that the Jacobson radical, all nilradicals, and the set of all nilpotents are equal in polynomial rings over symmetric-over-center rings. It is shown that reduced rings are reversible-over-center, and that given reversible-over-center rings, various sorts of reversible-over-center rings can be constructed. The structure of radicals in reversible-over-center and symmetric-over-center rings is also investigated.

• Journal of the Chungcheong Mathematical Society
• /
• v.27 no.1
• /
• pp.65-87
• /
• 2014

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. We show that if there exists a continuous linear Jordan derivation D : A ${\rightarrow}$ A such that [D(x), x]$D(x)^3{\in}$ rad(A) for all $x{\in}A$, then D(A) ${\subseteq}$ rad(A).

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.2
• /
• pp.211-224
• /
• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).

• Journal of the Korean Mathematical Society
• /
• v.49 no.3
• /
• pp.605-622
• /
• 2012

A ring $R$ is called semiregular if $R/J$ is regular and idem-potents lift modulo $J$, where $J$ denotes the Jacobson radical of $R$. We give some characterizations of rings $R$ such that idempotents lift modulo $J$, and $R/J$ satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) ${\pi}$-regular.

• Journal of the Korean Mathematical Society
• /
• v.54 no.1
• /
• pp.177-191
• /
• 2017

In this note we consider the right unit-duo ring property on the powers of elements, and introduce the concept of weakly right unit-duo ring. We investigate first the properties of weakly right unit-duo rings which are useful to the study of related studies. We observe next various kinds of relations and examples of weakly right unit-duo rings which do roles in ring theory.