• Communications of the Korean Mathematical Society
• /
• v.17 no.2
• /
• pp.221-227
• /
• 2002

In this paper we will show that the followings ; (1) Let R be a regular local ring of dimension n. Then $A_{n-2}$(R) = 0. (2) Let R be a regular local ring of dimension n and I be an ideal in R of height 3 such that R/I is a Gorenstein ring. Then [I] = 0 in $A_{n-3}$(R). (3) Let R = V[[ $X_1$, $X_2$, …, $X_{5}$ ]]/(p+ $X_1$$^{t1}$ + $X_2$$^{t2}$ + $X_3$$^{t3}$ + $X_4$$^2$+ $X_{5}$ $^2$/), where p $\neq$2, $t_1$, $t_2$, $t_3$ are arbitrary positive integers and V is a complete discrete valuation ring with (p) = mv. Assume that R/m is algebraically closed. Then all the Chow group for R is 0 except the last Chow group.group.oup.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.4
• /
• pp.797-800
• /
• 2008

Let R be a ring and I be a proper ideal of R. For the case of R being commutative, Anderson proved that (*) there are only finitely many prime ideals minimal over I whenever every prime ideal minimal over I is finitely generated. We in this note extend the class of rings that satisfies the condition (*) to noncommutative rings, so called homomorphically IFP, which is a generalization of commutative rings. As a corollary we obtain that there are only finitely many minimal prime ideals in the polynomial ring over R when every minimal prime ideal of a homomorphically IFP ring R is finitely generated.

• Bulletin of the Korean Mathematical Society
• /
• v.55 no.2
• /
• pp.625-631
• /
• 2018

Let R be a commutative ring with identity and let R[x] be the collection of polynomials with coefficients in R. There are a lot of multiplications in R[x] such that together with the usual addition, R[x] becomes a ring that contains R as a subring. These multiplications are from a class of functions ${\lambda}$ from ${\mathbb{N}}_0$ to ${\mathbb{N}}$. The trivial case when ${\lambda}(i)=1$ for all i gives the usual polynomial ring. Among nontrivial cases, there is an important one, namely, the case when ${\lambda}(i)=i!$ for all i. For this case, it gives the well-known Hurwitz polynomial ring $R_H[x]$. In this paper, we completely determine the Krull dimension of $R_H[x]$ when R is a $Pr{\ddot{u}}fer$ domain. Let R be a $Pr{\ddot{u}}fer$ domain. We show that dim $R_H[x]={\dim}\;R+1$ if R has characteristic zero and dim $R_H[x]={\dim}\;R$ otherwise.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.27 no.3
• /
• pp.349-356
• /
• 2003

The aim of this paper is to investigate the friction characteristics of piston assembly. The friction of piston assembly is composed of ring pack and skirt friction. In this paper, the theoretical models of piston ring pack and piston skirt were presented. The mixed lubrication theory was considered to calculate friction force of piston ring and skirt. from the results, most of friction in piston assembly occurred at the piston ring park. The piston assembly usually showed hydrodynamic lubrication characteristics. but the top and bottom dead centers showed mixed lubrication characteristics. The piston skirt was much affected by radial clearance and load, but ring was significantly influenced by ring tension.

• Journal of the Korean Mathematical Society
• /
• v.49 no.1
• /
• pp.85-98
• /
• 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(${\Gamma}_I(R)$). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ${\in}$ R, the vertices x and y are adjacent if and only if x + y ${\in}$ S(I). The total graph of a commutative ring, that denoted by T(${\Gamma}(R)$), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ${\in}$ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, $T({\Gamma}_I(R))=T({\Gamma}(R))$; this is an important result on the definition.

• Publications of The Korean Astronomical Society
• /
• v.30 no.2
• /
• pp.517-519
• /
• 2015

A large-scale neutral hydrogen ($H\small{I}$) ring serendipitously found in the Leo I galaxy group is 200 kpc in diameter with $M_{H\small{I}}{\sim}1.67{\times}10^9M_{\odot}$, unique in size in the Local Universe. It is still under debate where this $H\small{I}$ ring originated - whether it has formed out of the gas remaining after the formation of a galaxy group (primordial origin) or been stripped during galaxy-galaxy interactions (tidal origin). We are investigating the optical and $H\small{I}$ gas properties of the dwarf galaxies located within the gas ring in order to probe its formation mechanism. In this work, we present the photometric properties of the dwarfs inside the ring using the CFHT MegaCam $u^{\ast}$, $g^{\prime}$, $r^{\prime}$ and $i^{\prime}$-band data. We discuss the origin of the gas ring based on the stellar age and metal abundance of dwarf galaxies contained within it.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.6
• /
• pp.1401-1407
• /
• 2021

Let R be a commutative G-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal I of R is said to be graded radically principal if Grad(I) = Grad(〈c〉) for some homogeneous c ∈ R, where Grad(I) is the graded radical of I. The graded ring R is said to be graded radically principal if every graded ideal of R is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically principal if and only if every graded prime ideal is graded radically principal. Finally we study the graded radically principal property for the polynomial ring R[X].

• Communications of the Korean Mathematical Society
• /
• v.11 no.4
• /
• pp.887-893
• /
• 1996

Let (R,m) be a Cohen-Macaulay local ring with an infinite residue field and let $J = (a_1, \cdots, a_l)$ be a minimal reduction of an equimultiple ideal I of R. In this paper we shall prove that the following conditions are equivalent: (1) $I^2 = JI$. (2) $gr_I(R)/mgr_I(R)$ is Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal N. (3) $N^2 = (a'_1, \cdots, a'_l)N$, where $a'_i$ denotes the images of $a_i$ in I/mI for $i = 1, \cdots, l$.

• East Asian mathematical journal
• /
• v.36 no.3
• /
• pp.337-347
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.2
• /
• pp.555-565
• /
• 2014

An ideal I of a ring R is strongly ${\pi}$-regular if for any $x{\in}I$ there exist $n{\in}\mathbb{N}$ and $y{\in}I$ such that $x^n=x^{n+1}y$. We prove that every strongly ${\pi}$-regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any $x{\in}I$ there exist two distinct m, $n{\in}\mathbb{N}$ such that $x^m=x^n$. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly ${\pi}$-regular and for any $u{\in}U(I)$, $u^{-1}{\in}\mathbb{Z}[u]$.