• 호남수학학술지
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• 제23권1호
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• pp.5-14
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• 2001

Let E be an injective module over a commutative Noetherian ring A. In this paper we will show that if I is regular ideal, then the sequence of sets $$Ass_A((I^n)^{{\star}(E)}/I^n),\;n{\in}N$$ is ultimately constant. Also we obtain some related results. (Here for an ideal J of A, $J^{{\star}(E)}$ denotes the integral closure of J relative to E.

• Journal of applied mathematics & informatics
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• 제35권5_6호
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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.

• 대한수학회보
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• 제49권1호
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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.

• 대한수학회보
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• 제50권5호
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• pp.1711-1723
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• 2013

P. V$\acute{a}$mos called a ring R 2-good if every element is the sum of two units. The ring of all $n{\times}n$ matrices over an elementary divisor ring is 2-good. A (right) self-injective von Neumann regular ring is 2-good provided it has no 2-torsion. Some of the earlier results known to us about 2-good rings (although nobody so called at those times) were due to Ehrlich, Henriksen, Fisher, Snider, Rapharl and Badawi. We continue in this paper the study of 2-good rings by several authors. We give some examples of 2-good rings and their related properties. In particular, it is shown that if R is an exchange ring with Artinian primitive factors and 2 is a unit in R, then R is 2-good. We also investigate various kinds of extensions of 2-good rings, including the polynomial extension, Nagata extension and Dorroh extension.

• 대한수학회논문집
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• 제31권2호
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• pp.217-227
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• 2016

Let X be a nonempty set, and let $\mathfrak{F}=\{Y_i:i{\in}I\}$ be a family of nonempty subsets of X with the properties that $X={\bigcup}_{i{\in}I}Y_i$, and $Y_i{\cap}Y_j={\emptyset}$ for all $i,j{\in}I$ with $i{\neq}j$. Let ${\emptyset}{\neq}J{\subseteq}I$, and let $T^{(J)}_{\mathfrak{F}}(X)=\{{\alpha}{\in}T(X):{\forall}i{\in}I{\exists}_j{\in}J,Y_i{\alpha}{\subseteq}Y_j\}$. Then $T^{(J)}_{\mathfrak{F}}(X)$ is a subsemigroup of the semigroup $T(X,Y^{(J)})$ of functions on X having ranges contained in $Y^{(J)}$, where $Y^{(J)}:={\bigcup}_{i{\in}J}Y_i$. For each ${\alpha}{\in}T^{(J)}_{\mathfrak{F}}(X)$, let ${\chi}^{({\alpha})}:I{\rightarrow}J$ be defined by $i{\chi}^{({\alpha})}=j{\Leftrightarrow}Y_i{\alpha}{\subseteq}Y_j$. Next, we define two congruence relations ${\chi}$ and $\widetilde{\chi}$ on $T^{(J)}_{\mathfrak{F}}(X)$ as follows: $({\alpha},{\beta}){\in}{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}={\chi}^{({\beta})}$ and $({\alpha},{\beta}){\in}\widetilde{\chi}{\Leftrightarrow}{\chi}^{({\alpha})}{\mid}_J={\chi}^{({\alpha})}{\mid}_J$. We begin this paper by studying the regularity of the quotient semigroups $T^{(J)}_{\mathfrak{F}}(X)/{\chi}$ and $T^{(J)}_{\mathfrak{F}}(X)/{\widetilde{\chi}}$, and the semigroup $T^{(J)}_{\mathfrak{F}}(X)$. For each ${\alpha}{\in}T_{\mathfrak{F}}(X):=T^{(I)}_{\mathfrak{F}}(X)$, we see that the equivalence class [${\alpha}$] of ${\alpha}$ under ${\chi}$ is a subsemigroup of $T_{\mathfrak{F}}(X)$ if and only if ${\chi}^{({\alpha})}$ is an idempotent element in the full transformation semigroup T(I). Let $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ be the sets of functions in $T_{\mathfrak{F}}(X)$ such that ${\chi}^{({\alpha})}$ is injective, surjective and bijective respectively. We end this paper by investigating the regularity of the subsemigroups [${\alpha}$], $I_{\mathfrak{F}}(X)$, $S_{\mathfrak{F}}(X)$ and $B_{\mathfrak{F}}(X)$ of $T_{\mathfrak{F}}(X)$.