• Korean Journal of Mathematics
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• v.21 no.3
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• pp.319-323
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• 2013

Let D be a principal ideal domain, X be an indeterminate over D, D[X] be the polynomial ring over D, and $R_n=D[X]/(X^n)$ for an integer $n{\geq}1$. Clearly, $R_n$ is a commutative Noetherian ring with identity, and hence each nonzero nonunit of $R_n$ can be written as a finite product of irreducible elements. In this paper, we show that every irreducible element of $R_n$ is a primary element, and thus every nonunit element of $R_n$ can be written as a finite product of primary elements.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.645-648
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• 2005

Let K be a Galois extension of a field F with G = Gal(K/F). Let L be an extension of F such that $K\;{\otimes}_F\;L\;=\; N_1\;{\oplus}N_2\;{\oplus}{\cdots}{\oplus}N_k$ with corresponding primitive idempotents $e_1,\;e_2,{\cdots},e_k$, where Ni's are fields. Then G acts on $\{e_1,\;e_2,{\cdots},e_k\}$ transitively and $Gal(N_1/K)\;{\cong}\;\{\sigma\;{\in}\;G\;/\;{\sigma}(e_1)\;=\;e_1\}$. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T = $K\;{\otimes}_F\;R$, and suppose there are only finitely many prime ideals $Q_1,\;Q_2,{\cdots},Q_k$ of T with $Q_i\;{\cap}\;R\;=\;P$. Then G acts transitively on $\{Q_1,\;Q_2,{\cdots},Q_k\},\;and\;Gal(qf(T/Q_1)/qf(R/P))\;{\cong}\;\{\sigma{\in}\;G/\;{\sigma}-(Q_1)\;=\;Q_1\}$ where qf($T/Q_1$) is the quotient field of $T/Q_1$.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.1
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• pp.85-103
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• 2001

Suppose that Hand K are paired Hopf algebras and that A is an H - K - bicomodule algebra with multiplication which is a left H-comodule map and is a right K-comodule map. We define a new twisted algebra, $A^{\tau}$ and define $M^{\tau}$ for $M{\in}M_A^K$. We find an equivalent condition for $M^{\tau}{\in}M_{A^{\tau}}^K$. We show that the above defined twisted multiplication is the special case of Beattie's twist multiplication. We show that if K is commutative, then A is an H-module algebra and show that if $H^*$ is cocommutative then the construction of smash product appears as a special case of the new twist product.

• Korean Journal of Computational Design and Engineering
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• v.10 no.6
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• pp.432-443
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• 2005

Recent three-dimensional feature-based CAD systems based on solid or non-manifold modelling functionality have been widely used for product design in manufacturing companies. When product models associated with features are used in various downstream applications such as analysis, however, simplified and abstracted models at various levels of detail (LODs) are frequently more desirable and useful than the full detailed model. To provide multi-resolution models, the features need to be rearranged according to a criterion that measures the significance of the feature. However, if the features are rearranged, the resulting shape is possibly different from the original because union and subtraction Boolean operations are not commutative. To solve this problem, in this paper, the new concept of the effective zone of a feature is defined and identified using Boolean algebra. By introducing the effective zone, an arbitrary rearrangement of features becomes possible and arbitrary LOD criteria may be selected to suit various applications. Besides, because the effective zone of a feature is independent of the data structure of the model, the multi-resolution modelling algorithm based on the effective zone can be implemented on any 3D CAD system based on conventional solid representations as well as non-manifold topological (NMT) representations.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.463-471
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• 2013

Let R be a ring with identity 1, $I(R){\neq}\{0\}$ be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, $f{\in}I(R)$ ($e{\neq}f$), $e+f{\in}I(R)$. In this paper, the following are shown: (1) I(R) is a finite additive set if and only if $M(R){\backslash}\{0\}$ is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R; (2) for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite field).

• Journal of the Korean Mathematical Society
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• v.57 no.1
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• pp.21-59
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• 2020

Let L_K denote the hypothetical automorphic Langlands group of a number field K. In our recent study, we briefly introduced a certain unconditional non-commutative topological group ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$, called the Weil-Arthur idèle group of K, which, assuming the existence of L_K, comes equipped with a natural topological group homomorphism $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → L_K that we called the "Langlands form" of the global nonabelian norm-residue symbol of K. In this work, we present a detailed construction of ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ and $NR{\frac{\varphi}{K}^{Langlands}}$ : ${\mathfrak{W}}{\mathfrak{A}}{\frac{\varphi}{K}}$ → L_K, and discuss their basic properties.

• East Asian mathematical journal
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• v.14 no.2
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• pp.259-268
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• 1998

In this paper, for a commutative ring R with an identity, considering the endomorphism ring $End_R$(M) of left R-module $_RM$ which is (quasi-)injective or (quasi-)projective, some discriminations of prime endomorphism were found as follows: each epimorphism with the irreducible(or simple) kernel on a (quasi-)injective module and each monomorphism with maximal image on a (quasi-)projective module are prime. It was shown that for a field F, any given square matrix in $Mat_{n{\times}n}$(F) with maximal image and irreducible kernel is a prime matrix, furthermore, any given matrix in $Mat_{n{\times}n}$(F) for any field F can be factored into a product of prime matrices.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.195-204
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• 2021

Let R be a G-graded commutative ring with a nonzero unity and P be a proper graded ideal of R. Then P is said to be a graded uniformly pr-ideal of R if there exists n ∈ ℕ such that whenever a, b ∈ h(R) with ab ∈ P and Ann(a) = {0}, then bn ∈ P. The smallest such n is called the order of P and is denoted by ordR(P). In this article, we study the characterizations on this new class of graded ideals, and investigate the behaviour of graded uniformly pr-ideals in graded factor rings and in direct product of graded rings.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1387-1408
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• 2022

Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :_R M)∩S = ∅ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 ≠ am ∈ N, then either sa ∈ (N :_R M) or sm ∈ N. Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.83-91
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• 2023

A ring R is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal I of a UN ring R. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring R, a matrix ring M_n(R) is UN if and only if R is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group G and a field K for the group algebra KG to be UN. Then we extend the results obtained for KG to the group ring RG over a ring R (which may not necessarily be a field).