• 대한수학회보
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• 제54권6호
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• pp.1913-1925
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• 2017

Let R be a ring with identity. An ideal N of R is called ideal-symmetric (resp., ideal-reversible) if $ABC{\subseteq}N$ implies $ACB{\subseteq}N$ (resp., $AB{\subseteq}N$ implies $BA{\subseteq}N$) for any ideals A, B, C in R. A ring R is called ideal-symmetric if zero ideal of R is ideal-symmetric. Let S(R) (called the ideal-symmetric radical of R) be the intersection of all ideal-symmetric ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring R, we have $S(M_n(R))=M_n(S(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a quasi-Baer ring R, R is semiprime if and only if R is ideal-symmetric if and only if R is ideal-reversible.

• 대한수학회논문집
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• 제24권2호
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• pp.161-169
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• 2009

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.

• Korean Journal of Mathematics
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• 제6권2호
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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.

• 한국수학교육학회지시리즈A:수학교육
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• 제31권2호
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• pp.141-144
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• 1992

In this paper, some properties of p-injective ring is studied: The Jacobson radical of a pinjective ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Also, the left singular ideal of a ring which satisfies the ascending chain condition on essential left ideals is nilpotent. Finally, we give an example which shows that a semiprime left p-injective ring such that every essential left ideal is two-sided is not necessarily to be strongly regular.egular.

• 대한수학회논문집
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• 제13권1호
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• pp.1-6
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• 1998

The paper refers to ordered semigroups in which $x^2 (x \in S)$ are left ideal elements. We mainly show that this $po$-semigroup is left regular if and only if S is a union of left simple subsemigroups of S.

• 대한수학회보
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• 제56권1호
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• pp.23-29
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• 2019

Rings in which all accessible subrings are ideals are called filial. A ring R is called fully filial if all its subrings are filial (that is rings in which the relation of being an ideal is transitive). The present paper is devoted to the study of fully filial torsion rings. We prove a classification theorem for semiprime fully filial torsion rings.

• 충청수학회지
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• 제2권1호
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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.

• 대한수학회논문집
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• 제33권4호
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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.

• 대한수학회논문집
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• 제27권1호
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• pp.69-76
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• 2012

Let R be a ring, and ${\alpha}$ be an endomorphism of R. An additive mapping H : R ${\rightarrow}$ R is called a left ${\alpha}$-multiplier (centralizer) if H(xy) = H(x)${\alpha}$(y) holds for all x,y $\in$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left ${\alpha}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $H(x{\circ}y)-x{\circ}y=0$, (iv) $H(x{\circ}y)+x{\circ}y=0$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $H(x^2)=x^2$, (viii) $H(x^2)=-x^2$ for all x, y in some appropriate subset of R.

• 대한수학회보
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• 제31권2호
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• pp.163-165
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• 1994

Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.