• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.715-726
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• 2010

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs(Mat_n(R))\;=\;Mat_n(hs(R))$ for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.427-433
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• 2004

Conditions for a $\Gamma$-near-ring to be commutative are investigated.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.511-515
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• 2012

We first show that the semiprimeness, primeness, and reducedness can go up to up-monoid rings. By these results we can compute the lower nilradicals of up-monoid rings, from which the well-known fact of Amitsur and McCoy for the polynomial rings can be extended to up-monoid rings.

• East Asian mathematical journal
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• v.15 no.2
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• pp.165-176
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• 1999

After the derivation was defined in [19] by Posner a lot of researchers studied the derivations in ring theory in different manners such as in [2], [4], [5], ..., etc. Furthermore, many researches followed the definition of the generalized derivation([3], [6], [7], ..., etc.). Finally, Maksa defined a symmetric bi-derivation and many researches have been done in ring theory by using this definition. In this work, defining a symmetric bi-$\alpha$-derivation, we study the mentioned researches above in the light of this new concept.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.1-10
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• 2024

In this paper we will demonstrate some results on a prime ring with involution by introducing two generalized derivations acting on symmetric and skew symmetric elements. This approach allows us to generalize some well known results. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.27-39
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• 2021

We discuss the condition that every nonzero right annihilator of an element contains a nonzero ideal, as a generalization of the insertion-of-factors-property. A ring with such condition is called right AP. We prove that a ring R is right AP if and only if Dn(R) is right AP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. Properties of right AP rings are investigated in relation to nilradicals, prime factor rings and minimal order.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.819-826
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• 1998

The purpose of this paper is to prove the following results; (1) Let R be a prime ring of char $(R)\neq 2$ and I a nonzero left ideal of R. The existence of a nonzero symmetric bi-derivation D : $R\timesR\;\longrightarrow\;$ such that d is sew-commuting on I where d is the trace of D forces R to be commutative (2) Let m and n be integers with $m\;\neq\;0.\;or\;n\neq\;0$. Let R be a noncommutative prime ring of char$ (R))\neq \; 2-1\; p_1 \;n_1$ where p is a prime number which is a divisor of m, and I a nonzero two-sided ideal of R. Let $D_1$ ; $R\;\times\;R\;\longrightarrow\;and\;$ $D_2\;:\;R\;\times\;R\;longrightarrow\;R$ be symmetric bi-derivations. Suppose further that there exists a symmetric bi-additive mapping B ; $R\;\times\;R\;\longrightarrow\;and\;$ such that $md_1(\chi)\chi ＋ n\chi d_2(\chi)=f(\chi$) holds for all $\chi$$\in$I, where $d_1 \;and\; d_2$ are the traces of $D_1 \;and\; D_2$ respectively and f is the trace of B. Then we have $D_1=0 \;and\; D_2=0$.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.667-679
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• 2007

We study the prime ideal structure of the direct sum of directed system of rings indexed by a commutative semigroup which is nilpotent extension of a group.

• East Asian mathematical journal
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• v.27 no.1
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• pp.75-82
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• 2011

In this paper, we will introduce the concept of fuzzy mulitplication module. We will define a new operation called a product on th family of all fuzzy submodules of a fuzzy mulitplication module. We will define a fuzzy subset of the idealization ring R+M and find some relations with the product of fuzzy submodules and product of fuzzy ideals of the idealization ring R+M. Some properties of weakly fuzzy prime submoduels and fuzzy prime submodules which are de ned by T.K.Mukherjee M.K.Sen and D.Roy will be introduced. We will investigate some properties of fuzzy prime submodules of a fuzzy multiplication module.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.95-107
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• 2023

Consider ℛ to be an associative prime ring and 𝒦 to be a nonzero dense ideal of ℛ. A mapping (need not be additive) ℱ : ℛ → 𝒬_mr associated with derivation d : ℛ → ℛ is called a multiplicative b-generalized derivation if ℱ(αδ) = ℱ(α)δ +bαd(δ) holds for all α, δ ∈ ℛ and for any fixed (0 ≠)b ∈ 𝒬_s ⊆ 𝒬_mr. In this manuscript, we study the commutativity of prime rings when the map b-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative b-generalized derivation, which improves many results in the literature.