• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.711-720
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• 2021

Let R denote a commutative noetherian ring, and let 𝐱 := x1, …, xd be an R-regular sequence. Suppose that 𝖆 denotes a monomial ideal with respect to 𝐱. The first purpose of this article is to show that 𝖆 is irreducible if and only if 𝖆 is a generalized-parametric ideal. Next, it is shown that, for any integer n ≥ 1, (x1, …, xd)n = ⋂P(f), where the intersection (irredundant) is taken over all monomials f = xe11 ⋯ xedd such that deg(f) = n - 1 and P(f) := (xe1+11, ⋯, xed+1d). The second main result of this paper shows that if 𝖖 := (𝐱) is a prime ideal of R which is contained in the Jacobson radical of R and R is 𝖖-adically complete, then 𝖆 is a parameter ideal if and only if 𝖆 is a monomial irreducible ideal and Rad(𝖆) = 𝖖. In addition, if a is generated by monomials m1, …, mr, then Rad(𝖆), the radical of a, is also monomial and Rad(𝖆) = (ω1, …, ωr), where ωi = rad(mi) for all i = 1, …, r.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1319-1334
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• 2020

In this paper, we study the basic theory of the category of graded w-Noetherian modules over a graded ring R. Some elementary concepts, such as w-envelope of graded modules, graded w-Noetherian rings and so on, are introduced. It is shown that: (1) A graded domain R is graded w-Noetherian if and only if R^g_𝔪 is a graded Noetherian ring for any gr-maximal w-ideal m of R, and there are only finite numbers of gr-maximal w-ideals including a for any nonzero homogeneous element a. (2) Let R be a strongly graded ring. Then R is a graded w-Noetherian ring if and only if R_e is a w-Noetherian ring. (3) Let R be a graded w-Noetherian domain and let a ∈ R be a homogeneous element. Suppose 𝖕 is a minimal graded prime ideal of (a). Then the graded height of the graded prime ideal 𝖕 is at most 1.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.631-636
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• 2015

Let D be an integral domain and w be the so-called w-operation on D. In this note, we introduce the notion of *(w)-domains: D is a *(w)-domain if $(({\cap}(x_i))({\cap}(y_j)))_w={\cap}(x_iy_j)$ for all nonzero elements $x_1,{\ldots},x_m$; $y_1,{\ldots},y_n$ of D. We then show that D is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a *(w)-domain and $A^{-1}$ is of finite type for all nonzero finitely generated fractional ideals A of D.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.289-300
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• 2015

In [9], the author extends the definition of lifting and supplemented modules to ${\delta}$-lifting and ${\delta}$-supplemented by replacing "small submodule" with "${\delta}$-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of ${\delta}$-lifting and ${\delta}$-supplemented modules. Especially, we show that any finite direct sum of ${\delta}$-hollow modules is ${\delta}$-supplemented. On the other hand, the notion of amply ${\delta}$-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply ${\delta}$-supplemented and satisfies Descending Chain Condition (DCC) on ${\delta}$-supplemented modules and on ${\delta}$-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a ${\delta}$-semiperfect ring which satisfies DCC on ${\delta}$-small right ideals of R.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.603-613
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• 2006

Near-rings considered are right near-rings and R is a near-ring. $J_0^r(R)$, the right Jacobson radical of R of type-0, was introduced and studied by the present authors. In this paper $J_1^r(R)$ and $J_2^r(R)$, the right Jacobson radicals of R of type-1 and type-2 are introduced. It is proved that both $J_1^r$ and $J_2^r$ are radicals for near-rings and $J_0^r(R){\subseteq}J_1^r(R){\subseteq}J_2^r(R)$. Unlike the left Jacobson radical classes, the right Jacobson radical class of type-2 contains $M_0(G)$ for many of the finite groups G. Depending on the structure of G, $M_0(G)$ belongs to different right Jacobson radical classes of near-rings. Also unlike left Jacobson-type radicals, the constant part of R is contained in every right 1-modular (2-modular) right ideal of R. For any family of near-rings $R_i$, $i{\in}I$, $J_{\nu}^r({\oplus}_{i{\in}I}R_i)={\oplus}_{i{\in}I}J_{\nu}^r(R_i)$, ${\nu}{\in}\{1,2\}$. Moreover, under certain conditions, for an invariant subnear-ring S of a d.g. near-ring R it is shown that $J_2^r(S)=S{\cap}J_2^r(R)$.

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.121-126
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• 1985

D.K. Harrison [5] has shown that if R and S are fields of characteristic different from 2, then two Witt rings W(R) and W(S) are isomorphic if and only if W(R)/I(R)$^{3}$ and W(S)/I(S)$^{3}$ are isomorphic where I(R) and I(S) denote the fundamental ideals of W(R) and W(S) respectively. In [1], J.K. Arason and A. Pfister proved a corresponding result when the characteristics of R and S are 2, and, in [9], K.I. Mandelberg proved the result when R and S are commutative semi-local rings having 2 a unit. In this paper, we prove the result when R and S are 2-fold full rings. Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is nondegenerate (i.e., the natural mapping V.rarw.Ho $m_{R}$ (V, R) induced by B is an isomorphism), and with a quadratic mapping .phi.:V.rarw.R such that B(x,y)=(.phi.(x+y)-.phi.(x)-.phi.(y))/2 and .phi.(rx)= $r^{2}$.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U(R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis { $x_{1}$, .., $x_{n}$}, we will write < $a_{1}$,.., $a_{n}$> for (V, B, .phi.) where the $a_{i}$=.phi.( $x_{i}$) are in U(R), and denote the space by the table [ $a_{ij}$ ] where $a_{ij}$ =B( $x_{i}$, $x_{j}$). In the case n=2 and B( $x_{1}$, $x_{2}$)=1/2, we reserve the notation [ $a_{11}$, $a_{22}$] for the space.the space.e.e.e.

• Cross-Cultural Studies
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• v.40
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• pp.91-110
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• 2015

Both Plato and More imagined alternative ways of organizing society. What is common to both authors, then, is the fact that they resorted to fiction to discuss other options. They differed, however, in the way they presented that fiction. The concept of utopia is no doubt an attribute of modern thought, and one of its most visible consequences. But one of the main features of utopia as a literary genre is its relationship with reality. Utopists depart from the observation of the society they live in, note down the aspects that need to be changed and imagine a place where those problems have been solved. After the two World Wars, the twentieth century was predominantly characterized by man's disappointment at the perception of his own nature. In this context, utopian ideals seemed absurd and the floor was inevitably left to dystopian discourse. Both The Road by Cormac MacCarthy and The Childhood of Jesus by J. M. Coetzee can be called critical dystopia and critical utopia as they represent the imaginary place and time that author intended a contemporaneous reader to view as better or worse than contemporary society but with difficult problems that the described society may or may not be able to solve. As a changed adventure narrative, they have something in common like open ending, father and son relationship and religious allegory. But the most important thing is that they express the utopian impulse that is still energetic and transforming in the post-modern society.

• Membrane and Water Treatment
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• v.14 no.2
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• pp.65-76
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• 2023

Given the limited water resources and the presence of multiple decision makers with different and usually conflicting objectives in the exploitation of water resources systems, especially dam's reservoirs; therefore, the decision to determine the optimal allocation of reservoir water among decision-makers and stakeholders is a difficult task. In this study, by combining a fuzzy VIKOR technique or fuzzy multi-criteria decision making (FMCDM) and the Young's bilateral bargaining model, a new method was developed to determine the optimal quantitative and qualitative water allocation of dam's reservoir water with the aim of increasing the utility of decision makers and stakeholders and reducing the conflicts among them. In this study, by identifying the stakeholders involved in the exploitation of the dam reservoir and determining their utility, the optimal points on trade-off curve with quantitative and qualitative objectives presented by Mojarabi et al. (2019) were ranked based on the quantitative and qualitative criteria, and economic, social and environmental factors using the fuzzy VIKOR technique. In the proposed method, the weights of the criteria were determined by each decision maker using the entropy method. The results of a fuzzy decision-making method demonstrated that the Young's bilateral bargaining model was developed to determine the point agreed between the decisions makers on the trade-off curve. In the proposed method, (a) the opinions of decision makers and stakeholders were considered according to different criteria in the exploitation of the dam reservoir, (b) because the decision makers considered the different factors in addition to quantitative and qualitative criteria, they were willing to participate in bargaining and reconsider their ideals, (c) due to the use of a fuzzy-logic based decision-making approach and considering different criteria, the utility of all decision makers was close to each other and the scope of bargaining became smaller, leading to an increase in the possibility of reaching an agreement in a shorter time period using game theory and (d) all qualitative judgments without considering explicitness of the decision makers were applied to the model using the fuzzy logic. The results of using the proposed method for the optimal exploitation of Iran's 15-Khordad dam reservoir over a 30-year period (1968-1997) showed the possibility of the agreement on the water allocation of the monthly total dissolved solids (TDS)=1,490 mg/L considering the different factors based on the opinions of decision makers and reducing conflicts among them.

• Korean Journal of Heritage: History & Science
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• v.44 no.4
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• pp.172-205
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• 2011

Go bears significant meanings in terms of cultural and entertaining functions in Asia Eastern such as China and Japan. Beyond the mere entertaining level, it produces philosophical and mythic discourse as well. As a part of effort to seek an identity of Korean traditional garden culture, this study traced back to find meanings of rock-go-board and taste for the arts which ancestors pursued in playing Go game, through analysis and interpretation of correlation among origin of place name, nearby scenery, carved letters and vicinal handed-down place name. At the same time, their position, shape and location types were interpreted through comprehensive research and analysis of stone-go-boards including rock-go-board. Particularly, it focused on the rock names related to Sundoism(仙道) Ideal world, fixed due to a connection between traces of Sundoism and places in a folk etymology. Series of this work is to highlight features of the immortal sceneries, one of traditional landscaping ideals, by understanding place identity and scenic features of where the rock-go-boards are carved. These works are expected to become foundation for promotion and preservation of the traditional landscaping remains. The contents of this study could be summarized as follows; First, round stone and square board for round sky and angled land, black and white color for harmony of yin and yang and 361paths for rotating sky are symbols projecting order of universe. Sayings of Gyuljungjirak(橘中之樂), Sangsansaho(商山四皓), Nangagosa(爛柯故事) formed based on the idea of eternity stand for union of sky and sun. It indicates Go game which matches life and nature spatiotemporally and elegant taste for arts pursuing beauty and leisure. Second, the stone-go-boards found through this research, are 18 in total. 3 of those(16.1%), Gangjin Weolnamsaji, Yangsan Sohanjeong and Banryongdae ones were classified into movable Seokguk and 15(83.9%) including Banghakdong were turned out to be non-movable rock-go-boards carved on natural rocks. Third, upon the result of materializing location types of rock-go-boards, 15 are mountain stream type(83.9%) and 3 are rock peak type(16.1%). Among those, the one at Sobaeksam Sinseonbong is located at the highest place(1,389m). Considering the fact that all of 15 rock-go-boards were found at mountainous areas lower than 500m, it is recognizable that where the Go-boards are the parts of the living space, not far from secular world. Fourth, there are 7 Sunjang(巡將) Go with 17 Hwajeoms(花點), which is a traditional Go board type, but their existences, numbers and shapes of Hwajeom appear variously. Based on the fact, it is recognizable that culture of making go-board had been handed down for an extended period of time. Among the studied rock-goboards, the biggest one was Muju Sasunam[$80(82)cm{\times}80(82)cm$] while the smallest one was Yangsan Sohandjeong Seokguk ($40cm{\times}40cm$). The dimension of length and breadth are both $49cm{\times}48cm$ on average, which is realistic size for actual Go play. Fifth, the biggest bed rock, an under-masonry with carved Go-board on it, was one in Muju Sasunam[$8.7m{\times}7.5m(65.25m^2)$], followed by ones in Hoengseong Chuiseok[$7.8m{\times}6.3m(49.14m^2$] and Goisan Sungukam[$6.7m{\times}5.7m(37.14m^2)$]. Meanwhile, the smallest rock-go-board was turned out to be one in Seoul Banghak-dong. There was no consistency in directions of the Go-boards, which gives a hint that geographical features and sceneries of locations were considered first and then these were carved toward an optimal direction corresponding to the conditions. Sixth, rock-go-boards were all located in valleys and peaks of mountains with breathtaking scenery. It seems closely related to ancestors' taste for arts. Particularly, rock-go-boards are apprehended as facilities related to taste for arts for having leisure in many mountains and big streams under the idea of union of sky and human as a primitive communal line. Go became a medium of hermits, which is a traditional image of Go-game, and symbol of amusement and entertainment with the idea that Go is an essence of scholar culture enabling to reach the Tao of turning back to nature. Seventh, the further ancient time going back to, the more dreamlike the Go-boards are. It is an evident for that Sundoism, which used to be unacceptable once, became more visible and realistic. Considering the high relation between rock-go-boards and Sundoism relevant names such as Sundoism peak in Danyang Sobaeksan, 4 hermits rock in Muju and Sundoism hermit rock in Jangsu, Sundoism hermit rocks and rock-go-boards are sceneries and observation spots to express a communication of worship and longing for Sundoism. Eighth, 3 elements-physical environment such as location type of the rock-go-boards, human activities concentrated on 8 sceneries and Dongcheongugok(洞天九曲) setup and relevancy to Confucian scholars, as well as 'Sangsansaho' motif and 'Nangagosa' symbolic meaning were used as interpretation tools in order to judge the place identity. Upon the result, spatial investigation is required with respect to Sunyoodongcheon(仙遊洞天) concept based on enjoyment to unify with the nature rather than Dongcheongugok concept of neo-Confucian, for Dongcheon and Dongmoon(洞門) motives carved around the rock-go-boards. Generally, places where mountain stream type rock-go-boards were formed were hermit spaces of Confucianism or Sundoism. They are considered to have compromised one other with the change of times. Particularly, in the rock-go-board at the mountain peak, sublimity-oriented advent of Sundoism is considered as a significant factor to control place identity. Ninth, including where the rock-go-boards were established, the vicinal areas are well-known as parts of Dongcheongugok and Palkyung(八景) mostly. In addition, many of Sundoism relevant expressions were discovered even in the neighboring carvings written by scholars and nobility, which means sophisticated taste based on longing for Sundoism world played a significant role in making go-board. The rock-go-board is an integration of cultural phenomena naturally managed by seclusion of scholars in the Joseon Dynasty as well as remains and essence of Korean traditional landscaping. Some rock-go-boards out of 17 discovered in South Korea, including ones in Sobaeksan Sinsunbong, Banghak-dong, Chungju Gongili, Muju Sasunam, Yangsan Eogokdong Banryongdae Seokguk, are damaged such as cracks in rocks or fainted lines by hardships of time and hand stains. Worse yet, in case of Eunyang Bangudae Jipcheongjeong board, it is very difficult to identify the shape due to being buried. Rock-go-boards are valuable sculptures in terms of cultural asset and artwork since they reflect ancestors' love for nature and longing for Sundoism world. Therefore, they should be maintained properly with right preservation method. Not only rock-boards itself but also peripheral places are excellent cultural heritages and crucial cultural assets. In addition, vicinal sceneries of where rock-goboards and pavilion spots are the representative remains of embracing prototype of Korean traditional landscaping and major parts of cultural properties.