• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.451-456
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• 2009

Let R be a ring. We give some new characterizations of QF under the special annihilators condition. Some known results are obtained as corollaries.

• Korean Journal of Mathematics
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• v.18 no.3
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• pp.243-252
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• 2010

In this paper we extend the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left R-modules to the projective properties of representations of quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\rightarrow}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. And we show $0{\rightarrow}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module if and only if $0{\rightarrow}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. Then we show if P is a projective left R-module then $R[x]\longrightarrow^{id}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. We also show that if $L\longrightarrow^{id}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module, then $L[x]\longrightarrow^{id}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.527-534
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• 2005

We consider countable rings with ascending chain condition on right annihilators. We determine the structure of a countable right p-injective Baer ring, a countable semi prime quasi-Baer ring and a countable quasi-Baer biregular ring.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.1053-1057
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• 2020

We indicate that some results in [2] are wrong, and obtain some new results on them.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.109-126
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• 2012

In 2003, Marques-Smith and Sullivan described the join ${\Omega}$ of the 'natural order' $\leq$ and the 'containment order' $\subseteq$ on P(X), the semigroup under composition of all partial transformations of a set X. And, in 2004, Pinto and Sullivan described all automorphisms of PS(q), the partial Baer-Levi semigroup consisting of all injective ${\alpha}{\in}P(X)$ such that ${\mid}X{\backslash}X{\alpha}\mid=q$, where $N_0{\leq}q{\leq}{\mid}X{\mid}$. In this paper, we describe the group of automorphisms of R(q), the largest regular subsemigroup of PS(q). In 2010, we studied some properties of $\leq$ and $\subseteq$ on PS(q). Here, we characterize the meet and join under those orders for elements of R(q) and PS(q). In addition, since $\leq$ does not equal ${\Omega}$ on I(X), the symmetric inverse semigroup on X, we formulate an algebraic version of ${\Omega}$ on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.43-53
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• 1988

The main result of this paper is a characterization of almost projective modules over art in algebras by means of irreducible maps and almost split sequences. A module X is an almost projective module if and only if it has a presentation $0{\longrightarrow}L{\longrightarrow^{\alpha}}P{\longrightarrow}X{\longrightarrow}0$ with projective module P and irreducible maps ${\alpha}$. Let X be an injective almost projective non simple module and $0{\rightarrow}Dtr(x){\rightarrow}E{\rightarrow}X{\rightarrow}0$ be an almost split sequence. If $E=E_1{\oplus}E_2$ is a direct decomposition of indecomposable modules then ${\ell}(X)=3$.

• Honam Mathematical Journal
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• v.43 no.3
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• pp.403-416
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• 2021

It is known that the rank 2 stably free syzygy module P⁽²⁾ is not free. This algebraic fact was proved analytically, but this remarkable fact still lacks of a simple algebraic proof. The main purpose of this paper is to give a partially algebraic proof by making use of a theorem whose proof is quite topological, and the further properties of the module will be discussed.

• Honam Mathematical Journal
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• v.2 no.1
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• pp.13-18
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• 1980

모든 기호(記號)는 G.E Bredon의 저(著) Sheaf Theory의 기호(記號)를 따른다. A가 torsion free이며 elementary sheaf이라 하자. 그리고 L을 injective L-module이라 하자 $dim_{\varphi}X<{\infty}$이라면 support의 $family{\varphi}$와 locally subset z에 대하여 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{\varphi}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-p}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ 이며 support의 family c와 compact subset z에 대하여도 ${\Gamma}_{z}(^{\sim}Hom({\Gamma}_{c}(L),L){\otimes}A){\simeq}H_0{^{z}}(X:A)\;H_{-y}{^{z}}(X:A)=0,\;p=1,2,3,$⋯⋯ A가 elementary이면 locally closed z와 z에서 closed인 $z^{\prime}$ 그리고 $z^{\prime\prime}=z-z^{\prime}$에 대하여 exact sequence ⋯⋯${\rightarrow}H^{z^{\prime}}_{p}\;(X:A){\rightarrow}H^{z}_{p}(X:A){\rightarrow}H^{z^{\prime\prime}}_{p}\;(X:A){\rightarrow}$⋯⋯ 가 존재(存在)한다.

• Korean Journal of Mathematics
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• v.4 no.1
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• pp.45-49
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• 1996

Let $F_0$ be the maximal real subfield of $\mathbb{Q}({\zeta}_q+{\zeta}_q^{-1})$ and $F_{\infty}={\cup}_{n{\geq}0}F_n$ be its basic $\mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{\rightarrow}A_m$ induced by the inclusion $F_n{\rightarrow}F_m$ when $m>n{\geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{\rightarrow}A_m$ is injective for all $m>n{\geq}0$.

• Journal of applied mathematics & informatics
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• v.35 no.5_6
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• pp.565-586
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• 2017

In this paper we continue to investigate the Super Vertex Mean behaviour of all graphs up to order 5 and all regular graphs up to order 7. Let G(V,E) be a graph with p vertices and q edges. Let f be an injection from E to the set {1,2,3,${\cdots}$,p+q} that induces for each vertex v the label defined by the rule $f^v(v)=Round\;\left({\frac{{\Sigma}_{e{\in}E_v}\;f(e)}{d(v)}}\right)$, where $E_v$ denotes the set of edges in G that are incident at the vertex v, such that the set of all edge labels and the induced vertex labels is {1,2,3,${\cdots}$,p+q}. Such an injective function f is called a super vertex mean labeling of a graph G and G is called a Super Vertex Mean Graph.