• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.373-386
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• 2014

Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth($M/I^nM$) = d for $n{\gg}0$. Also we show that, if dim(R) = d and $I_1{\subset}\;{\cdots}\;{\subset}I_n$ is a chain of ideals of R such that $R/I_k$ is maximal Cohen-Macaulay for all k, then $n{\leq}{\ell}_R(R/(a_1,{\ldots},a_d)R)$ for every system of parameters $a1,{\ldots},a_d$ of R. Also, in the case where dim(R) = 2, we prove that the ideal transform $D_m(R/p)$ is minimax balanced big Cohen-Macaulay, for every $p{\in}Assh_R$(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.731-739
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• 2010

In this paper, we prove stability of the reciprocal difference functional equation $$r\(\frac{{\sum}_{i=1}^{m}x_i}{m}\)-r\(\sum_{i=1}^{m}x_i\)=\frac{(m-1){\prod}_{i=1}^{m}r(x_i)}{{\sum}_{i=1}^{m}{\prod}_{k{\neq}i,1{\leq}k{\leq}m}r(x_k)$$ and the reciprocal adjoint functional equation $$r\(\frac{{\sum}_{i=1}^{m}x_i}{m}\)+r\(\sum_{i=1}^{m}x_i\)=\frac{(m+1){\prod}_{i=1}^{m}r(x_i)}{{\sum}_{i=1}^{m}{\prod}_{k{\neq}i,1{\leq}k{\leq}m}r(x_k)$$ in m-variables. Stability of the reciprocal difference functional equation and the reciprocal adjoint functional equation in two variables were proved by K. Ravi, J. M. Rassias and B. V. Senthil Kumar [13]. We extend their result to m-variables in similar types.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.14 no.4
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• pp.941-951
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• 1994

The resilient behavior of subbase materials, locally used in Korea. were evaluated in this study by performing laboratory MR tests. The variations of the MR value according to LVDT's location in experiments and moisture conditions, respectively, were evaluated. And, in order to determine the relevant MR characteristics of the prototype materials, laboratory model tests containing smaller particles than those of the prototype were conducted. Based on above results, the relationship of the MR and the bulk stress (${\theta}$) was suggested. The case using internal LVDT. gave much larger $M_R$ value than that using external LVDT, since the external LVDT somewhat overestimates the resilient strain. The measured $M_R$ in damp conditions ($S_r$=40%) was larger than that in wet conditions ($S_r$=70%) by about 10%. The relationship between the $M_R$ and the void ratio was linear according to particle size effect. The relationship of the $M_R$ and the bulk stress (${\theta}$) on subbase materials in damp conditions to be used in Korea could be recommended as $M_R=3960{\cdot}{\theta}^{0.60}$ psi.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.573-583
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• 2005

In this paper, all near-rings R are left near-rings and all representations of R are (right) R-groups. We start with a study of AR, almost AR and AGR rings which are motivated by the works on the Sullivan's Problem [10] and its properties. Next, for any R-group G, we introduce a notion of R-groups with M R-property and investigate their properties and some characterizations of these R-groups. Finally, for the faithful M R-property, we get a commutativity of near-rings and rings.

• Journal of Korean Physical Therapy Science
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• v.9 no.2
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• pp.67-75
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• 2002

The purpose of this study is to know the average of hip joint range of motion and difference according to the aging for the elderly. This study consisted of elder male(n=75) and elder female(n=109). The result of assessment and analysis in hip pint range of motion are as follows : 1) The average hip flexion(knee flexed) joint range of motion in 60-69(from sixty to sixty-nine)years old are $104.26^{\circ}$(Left-Male), $101.00^{\circ}$(Right-Male), $107.05^{\circ}$(Left-Female), $107.05^{\circ}$(Right-Female). 70-79years old are $104.59^{\circ}$(L-M), $102.05^{\circ}$(R-M), $105.73^{\circ}$(L-F), $108.75^{\circ}$(R-F). 80-89years old are $101.53^{\circ}$(L-M), $101.13^{\circ}$(R-M), $96.83^{\circ}$(L-F), $97.67^{\circ}$(R-F). There was significant difference in hip flexion(knee flexed) among female group(p<.01). The average hip flexion(knee extended) joint range of motion in 60-69(from sixty to sixty-nine)years old are $73.13^{\circ}$(Left-Male), $72.04^{\circ}$(Right-Male), $77.29^{\circ}$(Left-Female), $75.97^{\circ}$(Right-Female). 70-79years old are $74.95^{\circ}$(L-M), $72.19^{\circ}$(R-M), $76.73^{\circ}$(L-F), $76.65^{\circ}$(R-F). 80-89years old are $70.83^{\circ}$(L-M), $70.37^{\circ}$(R-M), $69.00^{\circ}$(L-F), $69.00^{\circ}$(R-F). There was significant difference in left hip flexion(knee extended) among female group(p<.05). 2) The average hip extension joint range of motion in 60-69years old are $13.09^{\circ}$(L-M), $12.78^{\circ}$(R-M), $10.97^{\circ}$(L-F), $10.68^{\circ}$(R-F). 70-79years old are $8.95^{\circ}$(L-M), $8.48^{\circ}$(R-M), $11.24^{\circ}$(L-F), $10.90^{\circ}$(R-F). 80-89 years old are $8.40^{\circ}$(L-M), $8.23^{\circ}$(R-M), $7.33^{\circ}$(L-F), $7.33^{\circ}$(R-F). There was significant difference in left(p<.01) and right(p<.05) hip extension among male group(p<.05). 3) The average hip abduction joint range of motion in 60-69 years old are $33.04^{\circ}$(L-M), $33.17^{\circ}$(R-M), $33.16^{\circ}$(L-F), $33.37^{\circ}$(R-F). 70-79 years old are $31.00^{\circ}$(L-M), $30.05^{\circ}$(R-M), $32.44^{\circ}$(L-F), $32.68^{\circ}$(R-F). 80-89 years old are $29.07^{\circ}$(L-M), $27.90^{\circ}$(R-M), $28.17^{\circ}$(L-F), $28.67^{\circ}$(R-F). There was no significant difference among group. 4) The average hip adduction pint range, of motion in 60-69years old are $29.57^{\circ}$(L-M), $29.35^{\circ}$(R-M), $31.87^{\circ}$(L-F), $31.89^{\circ}$(R-F). 70-79, years old are $27.41^{\circ}$(L-M), 27.00(R-M) $30.85^{\circ}$(L-F), $31.28^{\circ}$(R-F). 80-89 years old are $26.87^{\circ}$(L-M), $26.63^{\circ}$(R-M), $24.67^{\circ}$(L-F), $24.83^{\circ}$(R-F). There was significant difference in hip abduction among female group(p<01). 5) The average hip external rotation pint range of motion in 60-69years old are $32.26^{\circ}$(L-M), $31.17^{\circ}$(R-M), $33.53^{\circ}$(L-F), $34.42^{\circ}$(R-F). 70-79 years old are $31.64^{\circ}$(L-M), $28.62^{\circ}$(R-M) $31.29^{\circ}$(L-F), $31.45^{\circ}$(R-F). 80-89 years old are $26.40^{\circ}$(L-M), $26.07^{\circ}$(R-M), $24.77^{\circ}$(L-F), $24.27^{\circ}$(R-F). There was significant difference in left(male, female p<.01) and right(female p<.0l) hip external rotation among group. 6) The average hip internal rotation joint range of motion in 60-69years old are $30.30^{\circ}$(L-M), $28.13^{\circ}$(R-M), $34.27^{\circ}$(L-F), $36.03^{\circ}$(R-F). 70-79years old are $31.24^{\circ}$(L-M), $29.57^{\circ}$(R-M), $28.51^{\circ}$(L-F), $29.10^{\circ}$(R-F). 80-89 years old are $24.63^{\circ}$(L-M), $24.40^{\circ}$(R-M), $24.27^{\circ}$(L-F), $24.27^{\circ}$(R-F). There was significant difference in left(male p<.05, female p<.01) and right(female p<.01) hip internal rotation among group.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.9 no.4
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• pp.281-284
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• 2009

In [8], we introduced the concept of fuzzy r-minimal structure which is an extension of smooth fuzzy topological spaces and fuzzy topological spaces in Chang's sense. And we also introduced and studied the fuzzy r-M continuity. In this paper, we introduce the concepts of fuzzy r-minimal compactness on fuzzy r-minimal compactness and nearly fuzzy r-minimal compactness, almost fuzzy r-minimal spaces and investigate the relationships between fuzzy r-M continuous mappings and such types of fuzzy r-minimal compactness.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.299-307
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• 2006

The following results ale extended from P-injective rings to AP-injective rings: (1) R is left self-injective regular if and only if R is a right (resp. left) AP-injective ring such that for every finitely generated left R-module M, $_R(M/Z(M))$ is projective, where Z(M) is the left singular submodule of $_{R}M$; (2) if R is a left nonsingular left AP-injective ring such that every maximal left ideal of R is either injective or a two-sided ideal of R, then R is either left self-injective regular or strongly regular. In addition, we answer a question of Roger Yue Chi Ming [13] in the positive. Let R be a ring whose every simple singular left R-module is Y J-injective. If R is a right MI-ring whose every essential right ideal is an essential left ideal, then R is a left and right self-injective regular, left and right V-ring of bounded index.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1221-1233
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• 2021

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.279-284
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• 1994

Let R = ($r_1$, $r_2$, …, $r_{m}$) and S = ($s_1$, $s_2$, …, $s_{n}$ ) be positive integral vectors satisfying $r_1$＋ $r_2$＋…＋ $r_{m}$ = $s_1$＋ $s_2$＋ㆍㆍㆍ＋ $s_{n}$ , and let U(R, S) denote the class of all m $\times$ n matrices A = [$_a{ij}$ ] where $a_{ij}$ = 0 or 1 such that (equation omitted) = $r_{i}$ , (equation omitted) = $s_{j}$ , i = 1, ㆍㆍㆍ, m, j = 1, ㆍㆍㆍ, n.(omitted)