• Honam Mathematical Journal
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• v.28 no.3
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• pp.327-332
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• 2006

Prime elements and ${\bigwedge}$-irreducible elements are introduced, and related properties are investigated.

• 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.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.1017-1023
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• 2010

We prove that a maximal ideal M of D[x] has two generators and is of the form where p is an irreducible element in a PID D having infinitely many nonassociate irreducible elements and q(x) is an irreducible non-constant polynomial in D[x]. Moreover, we find how minimal generators of maximal ideals of a polynomial ring D[x] over a DVR D consist of and how many generators those maximal ideals have.

• Journal of information and communication convergence engineering
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• v.8 no.3
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• pp.317-322
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• 2010

This paper presents an algorithm generating inverse element over finite fields GF($3^m$), and constructing method of inverse element generator based on inverse element generating algorithm. An inverse computing method of an element over GF($3^m$) which corresponds to a polynomial over GF($3^m$) with order less than equal to m-1. Here, the computation is based on multiplication, square and cube method derived from the mathematics properties over finite fields.

• Korean Journal of Mathematics
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• v.30 no.3
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• pp.425-431
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• 2022

Let ℤ and ℚ be the ring of integers and the field of rational numbers, respectively. Let h(ℤ, ℚ) be the ring of composite Hurwitz polynomials. In this paper, we study the factorization of composite Hurwitz polynomials in h(ℤ, ℚ). We show that every nonzero nonunit element of h(ℤ, ℚ) is a finite *-product of quasi-primary elements and irreducible elements of h(ℤ, ℚ). By using a relation between usual polynomials in ℚ[x] and composite Hurwitz polynomials in h(ℤ, ℚ), we also give a necessary and sufficient condition for composite Hurwitz polynomials of degree ≤ 3 in h(ℤ, ℚ) to be irreducible.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.513-519
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• 1997

We show that each orthomodular lattice containing only atomic nonpath-connected blocks is a full subalgebra of an irreducible path-connected orthomodular lattice and there is a path-connected orthomodualr lattice L containing a weakly path-connected full subalgebra C(x) for some element x in L.

• Proceedings of the IEEK Conference
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• 2004.08c
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• pp.514-518
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• 2004

This paper presents an algorithm generating inverse element over finite fields $GF(3^{m})$, and constructing method of inverse element generator based on inverse element generating algorithm. A method computing inverse of an element over $GF(3^{m})$ which corresponds to a polynomial over $GF(3^{m})$ with order less than equal to m-l. Here, the computation is based on multiplication, square and cube method derived from the mathematics properties over finite fields.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.503-508
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• 2007

Using the notion of posets, a method to make BCK-algebras is considered. We show that if a poset has the least element, then the induced BCK-algebra is bounded.

• Korean Journal of Mathematics
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• v.28 no.3
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• pp.439-447
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• 2020

Let M be a lattice module over a C-lattice L. Let Spec^s(M) be the collection of all second elements of M. In this paper, we consider a topology on Spec^s(M), called the second classical Zariski topology as a generalization of concepts in modules and investigate the interplay between the algebraic properties of a lattice module M and the topological properties of Spec^s(M). We investigate this topological space from the point of view of spectral spaces. We show that Spec^s(M) is always T₀-space and each finite irreducible closed subset of Spec^s(M) has a generic point.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.313-329
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• 2024

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ_≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.