• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.37-44
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• 2018

We prove some theorems showing that a right Jordan ideal or a left Jordan ideal of a 3-prime near-ring must be commutative if it admits a nonzero derivation acting as a homomorphism or an antihomomorphism. Moreover, we give examples proving necessity of the conditions given.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1097-1106
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• 2010

Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering $\Gamma$(R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of $\Gamma$(R) which is adjacent to every other vertex in $\Gamma$(R) if and only if R is a local ring or $R\;{\simeq}\;\mathbb{Z}_2\;{\times}\;F$ where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, $J^2$, $\ldots$, $J^n$, R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1085-1101
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• 2016

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.891-906
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• 2020

In this paper, the commutative module amenable Banach algebras are characterized. The hereditary and permanence properties of module amenability and the relations between module amenability of a Banach algebra and its ideals are explored. Analogous to the classical case of amenability, it is shown that the projective tensor product and direct sum of module amenable Banach algebras are again module amenable. By an application of Ryll-Nardzewski fixed point theorem, it is shown that for an inverse semigroup S, every module derivation of 𝑙¹(S) into a reflexive module is inner.

• Kyungpook Mathematical Journal
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• v.50 no.1
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• pp.1-5
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• 2010

Let P be a prime ideal of a commutative unital ring R; X an indeterminate; D := R/P; L the quotient field of D; F an algebraic closure of L; ${\alpha}$ ${\in}$ L[X] a monic irreducible polynomial; ${\xi}$ any root of in F; and Q = ${\alpha}$>, the upper to P with respect to ${\alpha}$. Then R[X]/Q is R-algebra isomorphic to $D[{\xi}]$; and is R-isomorphic to an overring of D if and only if deg(${\alpha}$) = 1.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1045-1056
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• 2020

Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, $^{\bar{\mathbb{AG}}}$(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n, m ∈ ℕ such that IⁿJ^m = (0) with Iⁿ, J^m ≠ (0). First, we differentiate when 𝔸𝔾(R) and $^{\bar{\mathbb{AG}}}$(R) coincide. Then, we have characterized the diameter and the girth of $^{\bar{\mathbb{AG}}}$(R) when R is a finite direct products of rings. Moreover, we show that $^{\bar{\mathbb{AG}}}$(R) contains a cycle, if $^{\bar{\mathbb{AG}}}$(R) ≠ 𝔸𝔾(R).

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.11-21
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• 2021

We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ≠ e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.327-339
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• 2023

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1031-1043
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• 2018

Let R be a commutative ring with $1{\neq}0$, I a proper ideal of R, and m and n positive integers. In this paper, we define I to be a weakly (m, n)-closed ideal if $0{\neq}x^m\;{\in}I$ for $x{\in}R$ implies $x^n{\in}I$, and R to be an (m, n)-von Neumann regular ring if for every $x{\in}R$, there is an $r{\in}R$ such that $x^mr=x^n$. A number of results concerning weakly(m, n)-closed ideals and (m, n)-von Neumann regular rings are given.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.289-298
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• 2017

Let R denote a commutative Noetherian (not necessarily local) ring and let I be an ideal of R of dimension one. The main purpose of this note is to show that the category ${\mathfrak{M}}(R,\;I)_{com}$ of I-cominimax R-modules forms an Abelian subcategory of the category of all R-modules. This assertion is a generalization of the main result of Melkersson in [15]. As an immediate consequence of this result we get some conditions for cominimaxness of local cohomology modules for ideals of dimension one. Finally, it is shown that the category ${\mathcal{C}}^1_B(R)$ of all R-modules of dimension at most one with finite Bass numbers forms an Abelian subcategory of the category of all R-modules.