• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.85-98
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• 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(${\Gamma}_I(R)$). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ${\in}$ R, the vertices x and y are adjacent if and only if x + y ${\in}$ S(I). The total graph of a commutative ring, that denoted by T(${\Gamma}(R)$), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ${\in}$ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, $T({\Gamma}_I(R))=T({\Gamma}(R))$; this is an important result on the definition.

• The Journal of the Korea institute of electronic communication sciences
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• v.13 no.4
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• pp.737-744
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• 2018

The RSA cryptosystem is a one of the first public-key cryptosystems and is widely used for secure data transmission and electric signature. The security of the RSA cryptosystem is based on the difficulty of factoring large numbers.. Though many studies on finding methods for factoring large numbers are going on, the results of that are all experimental or probabilistic. We, in this paper, construct an algorithm for finding large prime factors of integers without factoring integers using properties of the structure of semigroup of imaginary quadratic order and non-invertible ideal, then propose our methods foe deterministic attack against RSA cryptosystem.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1225-1236
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• 2016

Let R be a commutative ring with $1{\neq}0$ and n a positive integer. In this article, we introduce the n-Krull dimension of R, denoted $dim_n\;R$, which is the supremum of the lengths of chains of n-absorbing ideals of R. We study the n-Krull dimension in several classes of commutative rings. For example, the n-Krull dimension of an Artinian ring is finite for every positive integer n. In particular, if R is an Artinian ring with k maximal ideals and l(R) is the length of a composition series for R, then $dim_n\;R=l(R)-k$ for some positive integer n. It is proved that a Noetherian domain R is a Dedekind domain if and only if $dim_n\;R=n$ for every positive integer n if and only if $dim_2\;R=2$. It is shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by n-absorbing ideals for some n > 1.

• Cross-Cultural Studies
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• v.24
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• pp.19-31
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• 2011

This article examines the historical view of Benito $P{\acute{e}}rez$ $Gald{\acute{o}}s$ in his historical novel named "Zaragoza" and "$C{\acute{a}}novas$" by comparing with Numancia of Cervantes. In front of the national survival crisis, $Gald{\acute{o}}s$ recognizes that it is necessary that Spain should recover humanity and morality on basis of the krausist ideal. The krausists warned subjective idealism, stressing on free-will and moral conscience in a harmonious balance of rationality. Without refusing patriotism, they warned egoistic patriotism deformed by national selfishness or nationalism. Giner de los Ríos insisted on a necessity of harmonizing national inclination and universal one. $Gald{\acute{o}}s$ emphasizes that the identity of Spain is not possession of privileged class but sweat and tears of the mass of people in daily life. He represents Efemera as the ideal history on the basis of Unamuno's 'intrahistoria', by which, instead of $Luk{\acute{a}}cs^{\prime}$ progressive history, we can peep into a historical poetics of human free nature and vividness of discontinuous and endless shifting. According to the sub-history, we come to the conclusion that $Gald{\acute{o}}s$ and Cervantes would lay emphasis on sub-history, aiming to essence of Spanish soul originated from national landscape and people's daily life.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.163-169
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• 2011

Let D be an integral domain, S be a multiplicative subset of D such that DS is a PID, and D[X] be the polynomial ring over D. We show that S is an almost splitting set in D if and only if every nonzero prime ideal of D disjoint from S contains a primary element. We use this result to give a simple proof of the known result that D is a UMT-domain and Cl(D[X]) is torsion if and only if each upper to zero in D[X] contains a primary element.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.319-324
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• 2006

Let K be a finite cyclic extension of $k=\mathbb{F}_q(T)$ of prime degree ${\ell}$. Let ${\tilde{\mathcal{C}}}l_{K,{\ell}}$ be the Sylow ${\ell}$-subgroup of the ideal class group ${\tilde{\mathcal{C}}}l_K$ of $\mathcal{O}_K$. The structure of ${\tilde{\mathcal{C}}}l_{K,{\ell}}$ as $\mathbb{Z}_{\ell}[G]$/<$N_G$>-module is determined the dimensions $${\lambda}_i\;:=dim_{\mathbb{F}_{\ell}}({\tilde{\mathcal{C}}}l_{K,{\ell}}^{({\sigma}-1)^{i-1}}/{\tilde{\mathcal{C}}}l_{K,{\ell}}^{({\sigma}-1)^i})$$ for $i{\geq}1$. In this paper we investigate the dimensions ${\lambda}_1$ and ${\lambda}_2$.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.531-540
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• 2015

Let (R, m) be a Noetherian local ring, I an ideal of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{&gt;k}(I,A)$ the supremum of length of A-cosequence in dimension > k in I defined by Nhan-Hoang [8]. It is shown that for all $t{\leq}r$ the sets $$(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/I^n,A)))_{{\geq}k}\;and\\(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/(a_1^{n_1},{\cdots},a_l^{n_l}),A)))_{{\geq}k}$$ are independent of the choice of $n,n_1,{\cdots},n_l$ for any system of generators ($a_1,{\cdots},a_l$) of I.

• Journal of the Korean Mathematical Society
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• v.50 no.1
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• pp.219-229
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• 2013

Let R be a commutative ring with $1{\neq}0$ and let $\mathcal{H}$ = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If $R{\in}\mathcal{H}$, then R is called a ${\phi}$-ring. In this paper, we introduce the concepts of ${\phi}$-torsion modules, ${\phi}$-flat modules, and ${\phi}$-von Neumann regular rings.

• Journal of the Korean Mathematical Society
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• v.51 no.3
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• pp.593-607
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• 2014

Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.311-317
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• 2018

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.