• 대한수학회보
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• 제56권6호
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• pp.1385-1422
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• 2019

Let p be an odd prime. The algebraic structure of all negacyclic codes of length $8_{p^s}$ over the finite commutative chain ring ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ where $u^2=0$ is studied in this paper. Moreover, we classify all negacyclic codes of length $8_{p^s}$ over ${\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ into 5 cases, i.e., $p^m{\equiv}1$ (mod 16), $p^m{\equiv}3$, 11 (mod 16), $p^m{\equiv}5$, 13 (mod 16), $p^m{\equiv}7$, 15 (mod 16) and $p^m{\equiv}9$ (mod 16). From that, the structures of dual and some self-dual negacyclic codes and number of codewords of negacyclic codes are obtained.

• East Asian mathematical journal
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• 제36권3호
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• pp.337-347
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• 2020

Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a - aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.

• 대한화학회지
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• 제10권1호
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• pp.25-31
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• 1966

폴리아크릴로니트릴을 加熱할 때 着色과 同時에 構造變化가 있다는 것은 잘 알려진 事實이며 이것은 部分水素化나프틸리딘型環을 形成함에 基因하는 것으로 알려지고 있다. 本硏究는 이와 같은 構造變化의 反應을 動力學的 및 統計學的으로 取扱하여 지금까지 알려지지 않았던 새로운 事實들을 發見하였다. 첫째로 폴리아크릴로니트릴을 加熱할 때의 니트릴基의 減少는 一次反應이라는 것이며 이는 環形成反應이 긴連鎖反應으로 이루어지지 않는다는 것을 말해 주는 것이다. 다시 말하면 環形成時의 kinetic chain length는 極히 짧다는 것이다. 또 本 構造變化는 반드시 分子間反應(架橋結合)이 아니드라도, 일어날 수 있다는 것을 證明했다. 둘째로 環形成으로 니트릴基가 減少할 때 19~22%의 니트릴基가 殘存한다는 것이며 이를 統計學的으로 解析해본 結果 19.2%라는 計算値를 얻었으며 이는 實驗値와 잘 맞는 數値이다.

• 대한수학회보
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• 제37권1호
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• pp.191-207
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• 2000

Let R be a ommutative ring with unity and F a finite free R-module. For a nonnegative integer r, there exists a natural filtration of$S_r(S_2F)$ such that its associated graded module is isomorphic to $\Sigma_{{\lambda}{\epsilon}{\tau}_r}\;L_{\lambda}F$, where ${\Gamma}_{\gamma}$ set of partitions such that $$\mid${\lambda}$\mid$-2r,{{\widetilde}{\lambda}}-{{\widetilde}{\lambda}}_1},...,{{\widetilde}{\lambda}}_k},\;each\;{{\widetilde}{\lambda}}_t}$,is even. We call such filtrations plethysm formulas. We extend the above plethysm formula to the version of chain complexes. By plethysm formula we mean the composition of universally free functors. $Let{\emptyset}:G->F$ be a morphism of finite free R-modules. We construct the natural decomposition of $S_{r}(S_2{\emptyset})$,up to filtrations, whose associated graded complex is isomorphic to ${\Sigma}_{{\lambda}{\varepsilon}{\tau}}_r}\;L_{\lambda}{\emptyset}$.

• Kyungpook Mathematical Journal
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• 제55권2호
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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.

• 대한수학회보
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• 제55권4호
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• pp.1189-1208
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• 2018

The aim of this paper is to study the class of ${\Lambda}$-constacyclic codes of length $2p^s$ over the finite commutative chain ring ${\mathcal{R}}_a=\frac{{\mathbb{F}_{p^m}}[u]}{{\langle}u^a{\rangle}}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+{\cdots}+u^{a-1}{\mathbb{F}}_{p^m}$, for all units ${\Lambda}$ of ${\mathcal{R}}_a$ that have the form ${\Lambda}={\Lambda}_0+u{\Lambda}_1+{\cdots}+u^{a-1}{\Lambda}_{a-1}$, where ${\Lambda}_0,{\Lambda}_1,{\cdots},{\Lambda}_{a-1}{\in}{\mathbb{F}}_{p^m}$, ${\Lambda}_0{\neq}0$, ${\Lambda}_1{\neq}0$. The algebraic structure of all ${\Lambda}$-constacyclic codes of length $2p^s$ over ${\mathcal{R}}_a$ and their duals are established. As an application, this structure is used to determine the Rosenbloom-Tsfasman (RT) distance and weight distributions of all such codes. Among such constacyclic codes, the unique MDS code with respect to the RT distance is obtained.

• 대한수학회보
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• 제52권2호
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• pp.549-556
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• 2015

In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M, every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M, every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M, $End_RM$ is a division ring.

• 대한화학회지
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• 제37권1호
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• pp.62-67
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• 1993

Quinoline계 화합물인 미시적 비선형 광학 특성을 설명하기 위해 반경험적인 AMI 방법을 사용하여 초분극률을 계사하였다. Quinoline 고리에서 이차 비선형 특성에 기여하는 전자공여체, 아미노기와 전자수여체, 니트로기의 반응 위치를 변경하여 미시적 비선형 상수에 대한 이론적 영향을 조사해 보았다. 이차와 삼차 비선형 주쇄고분자로서 물리적, 기계적 성질이 우수한 polyquinoline계 고분자를 이용하기 이해 polyquinoline계 반복단위와 유사한 단분자 quinoline들을 설계하여 각각의 미시적 비선형 상수 및 바닥상태의 쌍극자 모멘트 등을 유한 전기장하에서 계산하여 서로 평가 비교해 보았다.