• 대한수학회논문집
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• 제30권4호
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• pp.363-377
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• 2015

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

• 대한수학회논문집
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• 제34권2호
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• pp.361-374
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• 2019

Let R be a ring, ($S,{\leq}$) a strictly ordered monoid, and ${\omega}:S{\rightarrow}End(R)$ a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring $R[[S,{\omega}]]$ is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and ${\omega}$ such that the skew generalized power series ring $R[[S,{\omega}]]$ is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.

• 한국수학교육학회지시리즈A:수학교육
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• 제12권2호
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• pp.5-12
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• 1974

단위원을 가지는 하환환에 있어서의 Prime Spectrum에 관하여 다음 세가지 사실을 증명하였다. 1. X를 환 R의 prime spectrum, C(X)를 X에서 정의되는 실연적함수의 환, X를 C(X)의 maximal spectrum이라 하면 X는 C(X)의 prime spectrum의 부분공간으로서의 한 T-space로 된다. N을 환 R의 nilradical이라 하면, R/N이 regula 이면 X와 X는 위상동형이다. 2. f: R$\longrightarrow$R'을 ring homomorphism, P를 R의 한 Prime ideal, $R_{p}$, R'$_{p}$를 각각 S=R-P 및 f(S)에 관한 분수환(ring of fraction)이라 하고, k(P)를 local ring $R_{p}$의 residue' field라 할 때, R'의 prime spectrum의 부분공간인 $f^{*-1}$(P)는 k(P)(equation omitted)$_{R}$R'의 prime spectrum과 위상동형이다. 단 f*는 f*(Q)=$f^{-1}$(Q)로서 정의되는 함수 s*:Spec(R')$\longrightarrow$Spec(R)이다. 3. X를 환 S의 prime spectrum, N을 R의 nilradical이라 할 때, 다음 네가지 사실은 동치이다. (1) R/N 은 regular 이다. (2) X는 Zarski topology에 관하여 Hausdorff 공간이다. (3) X에서의 Zarski topology와 constructible topology와는 일치한다. (4) R의 임의의 원소 f에 대하여 f를 포함하지 않는 R의 prime ideal 전체의 집합 $X_{f}$는 Zarski topology에 관하여 개집합인 동시에 폐집합이다.폐집합이다....

• 충청수학회지
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• 제19권4호
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• pp.349-356
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• 2006

The aim of this paper is to endow a monoid structure on the set S of all oriented knots(links) under the operation ${\biguplus}$, called addition of knots. Moreover, we prove that there exists a homomorphism of monoids between ($S_d,\;{\biguplus}$) to (N, +), where $S_d$ is a subset of S with an extra condition and N is the monoid of non negative integers under usual addition.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제26권4호
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• pp.233-245
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• 2019

Cyclic hypergroups are of great importance due to their applications to many field in mathematics. In this paper, we classify all polygroups of order less than six where each of its non-identity elements is a generator.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권1호
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• pp.73-88
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• 2004

A punctured torus $\Sigma(1,1)$ is a building block of oriented surfaces. The goal of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a punctured torus. Let $\cal{C}$ be a matrix presentation of the boundary component of $\Sigma(1,1)$.In the level of the matrix group $\mathbb{SL}$($\mathbb2,R$) we shall show that the trace of $\cal{C}$ is always negative.

• 대한수학회지
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• 제42권3호
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• pp.555-571
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• 2005

A pair of pants $\Sigma(0,3)$ is a building block of oriented surfaces. The purpose of this paper is to formulate the matrix presentations of elements of the Teichmuller space of a pair of pants. In the level of the matrix group $SL(2,\mathbb{R})$, we shall show that an odd number of traces of matrix presentations of the generators of the fundamental group of $\Sigma(0,3)$ should be negative.

• Kyungpook Mathematical Journal
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• 제62권4호
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• pp.641-655
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• 2022

A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

• 대한수학회보
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• 제56권1호
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• Kyungpook Mathematical Journal
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• 제51권1호
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• pp.93-108
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• 2011

A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from $Ra^n$ to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and $_RR$ is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.