• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.37-50
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• 2020

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. LCD cyclic codes have been known as reversible cyclic codes that had applications in data storage. Due to a newly discovered application in cryptography, interest in LCD codes has increased again. Although LCD codes over finite fields have been extensively studied so far, little work has been done on LCD codes over chain rings. In this paper, we are interested in structure of LCD codes over chain rings. We show that LCD codes over chain rings are free codes. We provide some necessary and sufficient conditions for an LCD code C over finite chain rings in terms of projections of linear codes. We also showed the existence of asymptotically good LCD codes over finite chain rings.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.2
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• pp.255-270
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• 2020

We studied the MDS self-dual codes over finite chain rings. We stated the projection and lifting of codes over the finite chain rings with respect to the MDS self-dual codes, and then we applied the results to the MDS self-dual codes over Galois rings.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.473-479
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• 2018

We investigate connections in the classes of rings with chain property and the lattice of strongly hereditary radicals.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1163-1173
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• 2021

In this paper, we obtain the MacWilliams identity for linear codes over finite chain rings with respect to homogeneous weight, and the product of chain rings.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1285-1295
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• 2015

An SM domain is an integral domain which satisfies the ascending chain condition on w-ideals. Then an SM domain also satisfies the descending chain condition on those chains of v-ideals whose intersection is not zero. In this paper, a study is begun to extend these properties to commutative rings with zero divisors. A $Q_0$-SM ring is defined to be a ring which satisfies the ascending chain condition on semiregular w-ideals and satisfies the descending chain condition on those chains of semiregular v-ideals whose intersection is semiregular. In this paper, some properties of $Q_0$-SM rings are discussed and examples are provided to show the difference between $Q_0$-SM rings and SM rings and the difference between $Q_0$-SM rings and $Q_0$-Mori rings.

• The Pure and Applied Mathematics
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• v.7 no.1
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• pp.33-40
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• 2000

We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1115-1123
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• 2016

Let $D{\subseteq}E$ be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D, E) and H(D, I) (resp., h(D, E) and h(D, I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D, E) satisfies the ascending chain condition on principal ideals if and only if h(D, E) satisfies the ascending chain condition on principal ideals, if and only if ${\bigcap}_{n{\geq}1}a_1{\cdots}a_nE=(0)$ for each infinite sequence $(a_n)_{n{\geq}1}$ consisting of nonzero nonunits of We also prove that H(D, I) satisfies the ascending chain condition on principal ideals if and only if h(D, I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1557-1565
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• 2022

In this paper, Glift codes, generalized lifted polynomials, matrices are introduced. The advantage of Glift code is "distance preserving" over the ring R. Then optimal codes can be obtained over the rings by using Glift codes and lifted polynomials. Zero divisors are classified to satisfy "distance preserving" for codes over non-chain rings. Moreover, Glift codes apply on MDS codes and MDS codes are obtained over the ring 𝓡 and the non-chain ring 𝓡_e,s.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.309-326
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• 2021

In near-ring theory several different types of primitivity exist. These all imply several different types of primeness. In case of near-rings with DCCN most of the types of primeness are known to imply primitivity of a certain kind. We are able to show that also so called 1-prime near-rings imply 1-primitivity. This enables us to classify maximal ideals in near-rings with chain condition with the concept of 1-primeness which leads to further results in the structure theory of near-rings.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.2011-2023
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• 2015

This paper determines the structures of one-homogeneous weight codes over finite chain rings and studies the algebraic properties of these codes. We present explicit constructions of one-homogeneous weight codes over finite chain rings. By taking advantage of the distance-preserving Gray map defined in [7] from the finite chain ring to its residue field, we obtain a family of optimal one-Hamming weight codes over the residue field. Further, we propose a generalized method that also includes the examples of optimal codes obtained by Shi et al. in [17].