• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.73-83
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• 2018

Let R be an associative ring with involution * and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an $({\alpha},{\beta})^*$-derivation of R if $d(xy)=d(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $F:R{\rightarrow}R$ is called a generalized $({\alpha},{\beta})^*$-derivation of R associated with an $({\alpha},{\beta})^*$-derivation d if $F(xy)=F(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [12], and a theorem of Daif and El-Sayiad [6], moreover, we generalize a theorem of Ali et al. [4] and a theorem of Huang and Koc [9] related to generalized Jordan triple $({\alpha},{\beta})^*$-derivations.

• Kyungpook Mathematical Journal
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• v.27 no.2
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• pp.191-195
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• 1987

In this note we introduce the concept of a lower radical for ${\Gamma}$-rings. As an application we also characterise the prime radical introduced by Barnes [1] as a lower radical. Furthermore it is shown that the prime radical can also be determined by the class of all semiprime ${\Gamma}$-rings.

• Journal of the Chungcheong Mathematical Society
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• v.2 no.1
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• pp.37-44
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• 1989

In this paper, it is shown that automatic continuity of derivations on some semi prime Banach algebras can be extended to higher derivations. In particular, we show that if every prime ideal is closed in a commutative semi prime Banach algebra then every higher derivation on it is continuous.

• Korean Journal of Mathematics
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• v.13 no.2
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• pp.217-221
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• 2005

Kehayopulu and Tsingelis [2] studied prime ideals of groupoids. Also the author studied prime left (right) ideals of groupoids. In this paper, we give some results on prime bi-ideals of groupoids.

• Journal for History of Mathematics
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• v.12 no.2
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• pp.151-158
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• 1999

In this paper, the concepts of semiprime $L$-fuzzy ideals and semiprimary $L$-fuzzy ideals of a $L$-fuzzy ring are introduced and some fundamental propositions proved. And we investigate the relation between fuzzy nil radical and semiprimary.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.3
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• pp.307-318
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• 2020

In this paper, we extend the notion of a generalized derivation F associated with derivation d to two generalized derivations F and G associated with the same derivation d, as a new idea, to obtain the commutativity of prime rings under certain conditions.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1271-1290
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• 2013

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1083-1096
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• 2018

Let R be an associative ring. We define a subset $S_R$ of R as $S_R=\{a{\in}R{\mid}aRa=(0)\}$ and call it the source of semiprimeness of R. We first examine some basic properties of the subset $S_R$ in any ring R, and then define the notions such as R being a ${\mid}S_R{\mid}$-reduced ring, a ${\mid}S_R{\mid}$-domain and a ${\mid}S_R{\mid}$-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite ${\mid}S_R{\mid}$-domain is necessarily unitary, and is in fact a ${\mid}S_R{\mid}$-division ring. However, we provide an example showing that a finite ${\mid}S_R{\mid}$-division ring does not need to be commutative. All possible values for characteristics of unitary ${\mid}S_R{\mid}$-reduced rings and ${\mid}S_R{\mid}$-domains are also determined.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.623-633
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• 2001

In this note we study the prime radicals of formal power series rings, and the shapes of them under the condition that the prime radical is nilpotent. Furthermore we observe the condition structurally, adding related examples to the situations that occur naturally in the process.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.149-154
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• 1998

We consider the ordered ${\Gamma}$-semigroups in which $x{\gamma}x(x{\in}M,{\gamma}{\in}{\Gamma})$ are left elements. We show that this $po-{\Gamma}$-semigroup is left regular if and only if M is a union of left simple sub-${\Gamma}$-semigroups of M.