• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1001-1017
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• 2023

Let R be a commutative graded ring with nonzero identity and n a positive integer. Our principal aim in this paper is to introduce and study the notions of graded n-irreducible and strongly graded n-irreducible ideals which are generalizations of n-irreducible and strongly n-irreducible ideals to the context of graded rings, respectively. A proper graded ideal I of R is called graded n-irreducible (respectively, strongly graded n-irreducible) if for each graded ideals I₁, . . . , I_n+1 of R, I = I₁ ∩ · · · ∩ I_n+1 (respectively, I₁ ∩ · · · ∩ I_n+1 ⊆ I ) implies that there are n of the I_i 's whose intersection is I (respectively, whose intersection is in I). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded n-irreducible ideal which is not an n-irreducible ideal and an example of a graded ideal which is graded n-irreducible, but not graded (n - 1)-irreducible.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.121-131
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• 2005

This paper is motivated by the results in [6]. We study some properties of strongly irreducible submodules of a module. In fact, our objective is to investigate strongly irreducible modules and to examine in particular when sub modules of a module are strongly irreducible. For example, we show that prime submodules of a multiplication module are strongly irreducible, and a characterization is given of a multiplication module over a Noetherian ring which contain a non-prime strongly irreducible submodule.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.263-272
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• 2011

In the paper [1], an explicit correspondence between certain cubic irreducible polynomials over $\mathbb{F}_q$ and cubic irreducible polynomials of special type over $\mathbb{F}_{q^2}$ was established. In this paper, we show that we can mimic such a correspondence for quintic polynomials. Our transformations are rather constructive so that it can be used to generate irreducible polynomials in one of the finite fields, by using certain irreducible polynomials given in the other field.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.189-196
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• 2021

Using ld-irreducible posets, we can easily characterize posets with respect to linear discrepancy. However, it is difficult to have the list of all the irreducible posets with respect to a given linear discrepancy. In this paper, we investigate some properties of disconnected posets and connected posets with respect to linear discrepancy, respectively and then we find various relationships between ld-irreducibily and connectedness. From these results, we suggest some methods to construct ld-irreducible posets.

• 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.

• Journal for History of Mathematics
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• v.18 no.1
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• pp.75-84
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• 2005

In this paper, we know the historical background in group representations and prove the properties such that a finite group G has non-trivial abelian normal subgroup in some condition for the irreducible character G and prove the properties of product of irreducible characters of finite groups.

• Honam Mathematical Journal
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• v.33 no.3
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• pp.355-366
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• 2011

In this paper we investigate criteria that for an irreducible monic quadratic polynomial f(x) ${\in}$ $\mathbb{Q}$[x], $f{\circ}g$ is reducible over $\mathbb{Q}$ for an irreducible polynomial g(x) ${\in}$ $\mathbb{Q}$[x]. Odoni intrigued the discussion about an explicit form of irreducible polynomials f(x) such that $f{\cric}f$ is reducible. We construct a system of infitely many such polynomials.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.177-182
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• 2003

A metacyclic group is a group that has a cyclic normal subgroup with cyclic quotient. We characterize finite metacyclic groups that have faithful irreducible representations over a given field.

• Journal for History of Mathematics
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• v.11 no.2
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• pp.83-90
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• 1998

Observing that for any compact space X, the minimal basically disconnected cover ${\bigwedge}Λ_X$ : ${\bigwedge}Λ_X{\leftrightarro}$ is ${\sigma}Z^#$-irreducible, we will show that the projective objects in the category of compact spaces and ${\sigma}Z^#$-irreducible maps are precisely basically disconnected spaces.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.38 no.1
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• pp.27-34
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• 2001

The multiplication algorithm using the primitive irreducible trinomial $x^m+x+1$ over $GF(2^m)$ was proposed by Mastrovito. The multiplier proposed in this paper consisted of the multiplicative operation unit, the primitive irreducible operation unit and mod operation unit. Among three units mentioned above, the Primitive irreducible operation was modified to primitive irreducible trinomial $x^m+x+1$ that satisfies the range of 1$x^m,{\cdots},x^{2m-2}\;to\;x^{m-1},{\cdots},x^0$ is reduced. In this paper, the primitive irreducible polynomial was reduced to the primitive irreducible trinomial proposed. As a result of this reduction, the primitive irreducible trinomial reduced the size of circuit. In addition, the proposed design of multiplier was suitable for VLSI implementation because the circuit became regular and modular in structure, and required simple control signal.